Trace theories, Bokstedt periodicity and Bott periodicity
aa r X i v : . [ m a t h . K T ] A p r Trace theories, B¨okstedt periodicity and Bottperiodicity
D. Kaledin ∗ Contents ∗ Supported by the Russian Science Foundation, grant 18-11-00141. Examples. 67 -categories. 83
10 Homotopy trace theories. 168
11 Topological Hochschild Homology. 186
Introduction.
The big miracle of Topological Hochschild Homology is that it is so small: if k is a perfect field of some positive characteristic p , then T HH q ( k ) = k [ σ ],the algebra of polynomials in one variable σ of homological degree 2 knownas the B¨okstedt periodicity element . If one only cares about
T HH q ( k ) asa graded k -vector space, then there is a spectral sequence converging to itwhose E -page is easy to compute. The sequence degenerates at E p , so thatif p = 2, it just degenerates (but one still has to do a separate argument ifone is interested in the multiplicative structure). For odd p , the sequencedoes not degenerate: the E -page is rather large, but the differential d p − is non-trivial and cuts it down to its proper size. Again, the multiplicativestructure has to be studied separately.This miraculous behavior of T HH can be proved in several different ways(we give a brief overview below in Subsections 11.2, 11.4). However, noneof the proofs are easy. This paper arose as an attempt to answer a questionof L. Hesselholt: is there now, when Topological Hochschild Homology is inits fourth decade, a simple and/or conceptually clear proof?Let us state right away that our attempt mostly failed. The best we cando with the spectral sequence argument is a modern repackaging of the 1994proof of [FLS]. We add strictly polynomial functors that were not available3t the time. The resulting argument is reasonably short but needs one exter-nal input, namely, the fact that the category of strictly polynomial functorsof some fixed degree has finite homological dimension. For multiplication,we can do slightly better: we construct a ring map ϕ from T HH q ( k ) to somering such that ϕ ( σ ) is not nilpotent; this means that k [ σ ] is a subalgebra in T HH q ( k ), and then it must be the whole thing for dimension reasons. Themap ϕ can be obtained by a certain truncation of the cyclotomic structuremap of [NS], but it is actually simpler to construct it by general nonsense.To compensate for our failure at the main stated goal, we also prove twocomparison theorem. The first one gives a really simple and purely alge-braic expression of the Topological Hohchschild Homology T HH q ( A, M ) ofa k -algebra A with coefficients in an A -bimodule M in terms of the so-calledHochschild-Witt Homology W HH q ( A, M ) introduced in [Ka8], [Ka11] (seealso an overview in [Ka10]). If M = A is the diagonal bimodule, then wealso analyze the circle action on T HH q ( A ) = T HH q ( A, A ), and prove thatthe periodic Topological Cyclic Homology
T P q ( A ) of [H4] coincides with theperiodic version W HP q ( A ) of W HH q ( A ). The second comparison theoremconcerns only the diagonal bimodule case: we prove that T HH q ( A ) coincideswith zero term of the “conjugate filtration” on the co-periodic cyclic homol-ogy CP q ( A ) introduced in [Ka7]. In particular, if one inverts the B¨okstedtgenerator σ , then T HH ( A ) becomes CP q ( A ), and the identification sends σ to the Bott periodicity generator u − .All of the above is contained in Section 11, and at this point, the readermight wonder what are we doing in the first 10 sections, and indeed why isthe paper so long. The answer to this is technological.Our main technical tool for both comparison theorems is the notion of a“trace theory” sketched in [Ka3]. Roughly speaking, the idea is to observethat T HH q ( A, M ) for different A and M are related by certain canonicalisomorphisms, and one can use it to show that T HH q ( k, M ), for any M ,completely determines T HH q ( A, M ) for any A and M . In other words, ifone works with arbitrary coefficients, and takes account of the trace theorystructure, then one can forget about arbitrary k -algebras and only consider k itself. A toy version of such a reconstruction theorem was proved in [Ka3].However, doing the whole theory properly and in a reasonable generalityrequires quite a lot of work.The second tool that we use heavily throughout the paper is the tech-nique of “stabilization”, or “additivization”. Namely, for any functor F ( M )from some additive category such as that of bimodules to the category ofspaces, one can take its Goodwillie derivative, thereby forcing the functor toland in connective spectra and become additive with respect to M . We have4ound out that this works really well for T HH q ( A, M ) and related theories:both the theories and relevant maps between them such as our map ϕ canbe obtained by stabilization from something really simple and obvious (for T HH q ( A, M ), this is the “cyclic nerve” of [Go]). Informally, it pays to onlygo stable at the very end. However, once again, developing the stabilizationmachine that is strong enough and versatile enough requires some space.The two stories, that of stabilization and that of trace theories, arebasicaly independent, so that the structure of the paper is non-linear. Arough leitfaden would look something like this:Section 1,2,3 −−−−→
Section 4,5 y y
Section 6,7,8,9 −−−−→
Section 10,11 . Before we explain in more detail what is being done where, let us make acomment on methodology.The natural context for trace theories is that of 2-categories: a tracetheory is defined on a 2-category, and a related notion of a “trace functor”makes sense for monoidal categories (that is, 2-categories with one object).The theory of 2-categories is notorious for horrible multidimensional dia-grams, “higher associativity constraints” and the like, with proofs beingeither impossibly long, or incomplete. In our experience, the only way toavoid this is to use systematically the usual category theory and the machin-ery of Grothendieck fibrations. We thus replace a 2-category with its nerve,and treat it as a usual category fibered over ∆, modulo some conditions(but no extra structure). Functors between 2-categories are then functorsover ∆, again modulo some conditions, and they are always constructedby adjunction — typically, as left or right Kan extensions. All the “higherconstraints” remain packaged via the Grothendieck construction and do notappear explicitly; one never has to check that two particular morphismscoincide.For better or for worse, our approach to things homotopical is thenexactly the same: we rely on the usual category theory and adjunction. Weassume known that for any small category I , localizing functors from I tospaces gives a well-defined category Ho( I ), and for any functor γ : I ′ → I ,the pullback functor γ ∗ : Ho( I ) → Ho( I ′ ) has a left and right-adjoint givenby homotopy Kan extensions γ ! , γ ∗ : Ho( I ′ ) → Ho( I ). These facts canbe established in any way one likes (for example, by using Reedy modelstructures, as in [DHKS]), and then used as a black box. Everything else5roceeds by using Kan extensions, and the formalism is often identically thesame as in the non-homotopical setting.This naive approach to homotopy theory has some obvious disadvantages— for example, we cannot works with E ∞ -ring spectra, nor with equivariantand cyclotomic spectra — but for our purposes in the paper, it is sufficient.Let us now give a more detailed overview of the paper. Sections 1, 2and 3 are preliminary. Section 1 is concerned with category theory — es-sentially, glorified bookkeeping — so that nothing whatsoever is new, andthe main goal is to fix notation and terminology (one slightly non-standardthing is the notion of a “framing” in Lemma 1.14 that turns out to bevery convenient for computing Kan extensions). Section 2 deals with theGrothendieck construction and Grothendieck fibrations. In fact, since wehave no need of model categories anywhere in the paper, we take the lib-erty of restoring the original terminology of [Gr] and speak about fibrationsand cofibrations, with cofibrations probably more important (for reasonsexplained in Subsection 6.4). Subsections 2.1 and 2.2 are fairly standard,Subsection 2.3 contains a version of the right Kan extension constructionfor cofibrations, and Subsection 2.4 contains technical material that shouldprobably be skipped until needed (“kernels” of Definition 2.16 are only usedin Subsection 7.5, and “reflections” are only used to construct the relativefunctor categories (2.45) and (2.47) needed in Subsection 4.4). Section 3 ispure combinatorics: we fix notation for the category ∆ of finite non-emptyordinals and related small categories such as the cyclic category Λ. Again,nothing is new (the treatment of Λ follows [Ka11, Subsection 1.2]).Section 4 starts the homotopical part of the story. First, we recall thestandard Segal machine of [S2], and we show that with very small modifica-tion, the same arguments provide a stabilization functor for functors fromsome pointed category with finite coproducts to spaces. We then get someadditional results by purely formal games with adjunction, and we treatmultiplicative structures, too (in a very naive way, see Remark 4.24). InSection 5, we compute some examples of stabilization; on one hand, thisillustrates the general machinery, and on the other hand, provides the maintechnical results needed for Section 11. In particular, we show how one canobtain the dual Steenrod algebra by stabilization, and we also show thatit has a natural filtered counterpart obtained by stabilizing divided powerfunctors. This possibly extends all the way to a filtered version of the stablehomotopy category, see Remark 5.15, but we do not explore this seriouslyat this point. The main technical tool is the notion of “truncated Tate co-homology” of a finite group with respect to a family of subgroups; this is6he only thing that might possibly be new (at least we do not know a readyreference).In Section 6, we temporarily abandon homotopy theory and move to2-categories. Monoidal structures and 2-categories are treated in Section6 itself. Section 7 is devoted to adjunction and adjoint pairs in general 2-categories, and some results here might be new – in particular, this concernsa surprisingly simple description of a free 2-category generated by an n -tupleof composable adjoint pairs given in Subsection 7.5. We should mentionthough that this description is only needed in Subsection 9.2, so it can beskipped at first reading. Moreover, the somewhat technical Proposition 7.9can be used as a black box. In Section 8, we construct and study one specific2-category M or ∗ ( C ) of small categories enriched over a fixed monoidal cat-egory C . In Section 9, we introduce trace theories, and we prove our mainreconstruction theorem: a trace theory on M or ∗ ( C ) is completely deter-mined by the associated trace functor on C (the actual statement is slightlystronger and more precise, see Theorem 9.10).Sections 10 and 11 bring the two stories together. Section 10 is a ho-motopical version of Section 9, and the gist of it is that everything donefor trace theories extends to homotopy trace theories verbatim , simply byadding the adjective “homotopy” in appropriate places. We also combinethis with Section 4, by introducing the notion of a “stable homotopy tracetheory” and showing how to do stabilization in this context. With all thisout of the way, we reap the benefits: in Section 11, we apply the machineryto THH and prove all the theorems mentioned above. We distinguish between large and small categories butavoid more advanced set theory. We denote by Sets the category of sets, andwe denote by Cat the category of small categories. For any category C , wewrite c ∈ C as a shorthand for “ c is an object in C ”, and for any c, c ′ ∈ C , wewrite C ( c, c ′ ) for the set of maps from c to c ′ (we recall that by definition, thisis always a set). We denote by C o the opposite category, C o ( c ′ , c ) = C ( c, c ′ ),and for any morphism f ∈ C ( c, c ′ ), we denote by f o ∈ C o ( c ′ , c ) = C ( c, c ′ )the corresponding morphism in the opposite category. We let C > resp. C < be the category obtained by adding a new terminal resp. initial object o to the category C , and we note that we have C o< ∼ = C >o , C o> ∼ = C
Iso of all invertible maps. A functor γ : C ′ → C from a groupoid C ′ by definition factors uniquely through C Iso ,and if C ′ is strongly discrete, it factors uniquely through C Id . We denoteby ∀ the class of all maps in C , so that C ∀ = C . As usual, a functor γ is conservative if γ ∗ Iso = Iso .If C ′ is discrete, then any functor γ : C ′ → C is isomorphic to a functorthat factors through C Id but unfortunately, this factorization is not unique(any two are isomorphic as functors to C but not to C Id ). To aleviate theproblem, we say that a category C is tight if it has exactly one object in eachisomorphism class, we say that a tighetening of some C is a tight weakly fullsubcategory C ′ ⊂ C such that the embedding C ′ → C is an equivalence, andwe assume that tightenings exist (this requires slightly more of the axiomof choice that one would like but that’s life). If C is tight, then any functor C ′ → C from a discrete C factors uniquely through C Id , and if C itself isdiscrete, it is strongly discrete.For any category I , its category of arrows ar ( I ) has arrows f : i ′ → i asobjects, with morphisms from f : i ′ → i to f : i ′ → i given by commutativediagrams(1.3) i ′ f −−−−→ i g ′ y y g i ′ f −−−−→ i . The twisted arrow category tw ( I ) has the same objects as ar ( I ), but mor-phisms are given by commutative diagrams(1.4) i ′ f −−−−→ i g ′ x y g i ′ f −−−−→ i . For any dense subcategory I v ⊂ I defined by a class of morphisms v , wedenote by ar v ( I ) ⊂ ar ( I ), tw v ( I ) ⊂ tw ( I ) the full subcategories spanned byarrows in v . Sending an arrow i ′ → i to its target i gives functors(1.5) t : ar ( I ) → I, t : tw ( I ) → I, and sending it to the source i ′ gives functors(1.6) s : ar ( I ) → I, s : tw ( I ) → I o .
9e also have a functor(1.7) η : I → ar ( I )sending an object i ∈ I to id : i → i , and η is fully faithful, right-adjoint to t and left-adjoint to s . Definition 1.1. A factorization system on a category I is a pair of classes h L, R i defining dense subcategories I L , I R ⊂ I such that L, R ⊃ Iso , anymorphism f : i ′ → i in I admits a factorization(1.8) i ′ l −−−−→ i ′′ r −−−−→ i with l ∈ L , r ∈ R , and for any commutative square i f −−−−→ i ′ l y y r i g −−−−→ i ′ in I with l ∈ L , r ∈ R there exists a unique morphism q : i → i ′ such that f = q ◦ l and g = r ◦ q . Example 1.2. If I = I × I is the product of two categories I , I , withthe projections π l : I → I l , l = 0 ,
1, then the classes π ∗ Iso and π ∗ Iso form afactorization system on I (in either order). Example 1.3.
A factorization system h L, R i on a category I defines a fac-torization system h ( s × t ) ∗ ( L × L ) , ( s × t ) ∗ ( R × R ) i on the arrow category ar ( I ) that restricts to factorization systems on ar L ( I ) , ar R ( I ) ⊂ ar ( I ). Wehave ar L ( I ) ( s × t ) ∗ ( L × L ) ∼ = ar ( I L ) and ar R ( I ) ( s × t ) ∗ ( R × R ) ∼ = ar ( I R ).For further details about factorization systems, we refer the reader to[Bou]. In particular, Definition 1.1 implies that L ∩ R = Iso , either of theclasses L , R competely determines the other one, and the factorization (1.8)is unique up to a unique isomorphism. Definition 1.4.
A full subcategory I ′ ⊂ I is left resp. right-admissible if theembedding functor γ : I ′ → I admits a left resp. right-adjoint γ † : I → I ′ . Example 1.5.
If we have a left resp. right-admissible full subcategory I ′ ⊂ I , and another subcategory I ′′ ⊂ I that contains I ′ and the adjunction map i → γ ( γ † ( i )) resp. γ ( γ † ( i )) → I for any i ∈ I ′′ , then I ′ ⊂ I ′′ is also left resp.right-admissible, with the same γ † . 10 xample 1.6. Assume given a category I equipped with a factorizationsystem h L, R i , and for any i ∈ I , denote by I/ R i ⊂ I/i the full subcat-egory spanned by pairs h i ′ , α ( i ′ ) i with α ( i ′ ) ∈ R . Then I/ R i ⊂ I/i isleft-admissible, with the adjoint functor sending an arrow to the R -part ofits decomposition (1.8), and so is I L / R i ⊂ I L /i . For any two categories C , C ′ with C small, functors from C to C ′ form a well-defined category that we denote by Fun( C , C ′ ) (for example,if C = [1] is the single arrow category with two objects 0, 1 and a single non-identity map 0 →
1, then Fun([1] , C ′ ) ∼ = ar ( C ′ ) is the arrow category ofthe category C ′ ). If C is not small, one says that a functor γ : C → C ′ is continuous if it commutes with filtered colimits, and one says that a category C is finitely presentable if it has all filtered colimits and a set C of compactobjects such that any c ∈ C is a filtered colimit of objects in C . Then if C islarge but finitely presentable, continuous functors from C to C ′ also form awell-defined category that we also denote by Fun( C , C ′ ). For brevity, we willsay that C is bounded if it is either small or large and finitely presentable,and we will say that a functor γ : C → C ′ is bounded if either C is small or C is large and γ is continuous, so that in either case, Fun( C , C ′ ) denotes thecategory of bounded functors. Whenever we consider a functor γ : C → C ′ with bounded C , it is assumed to be bounded unless indicated otherwise.We have the evaluation functor (1.9) ev : C ×
Fun( C , C ′ ) → C ′ with the obvious universal property: for any bounded category C ′′ andbounded functor γ : C × C ′′ → C ′ , there exists a bounded functor γ ′ : C ′′ → Fun( C , C ′ ) and an isomorphism α : γ ∼ = ev ◦ ( Id × γ ′ ), and the pair h γ ′ , α i isunique up to a unique isomorphism.We say that a functor γ : C → C ′ inverts a map f if γ ( f ) is invertible, andwe say that γ is locally constant if it inverts all maps, and constant if up to anisomorphism, it factors through the tautological projection C → pt . For anyclass of maps v in a bounded C , we denote by Fun v ( C , C ′ ) ⊂ Fun( C , C ′ ) thefull subcategory spanned by functors that invert maps in v . In particular,Fun ∀ ( C , C ′ ) is the category of locally constant functors. For any subclass w ⊂ v , we have Fun v ( C , C ′ ) ⊂ Fun w ( C , C ′ ), and we say that w is dense in v if the inclusion is an equivalence for any C ′ .For any bounded categories I , I ′ and any category E , a bounded functor γ : I → I ′ induces a pullback functor γ ∗ : Fun( I ′ , E ) → Fun( I, E ), F F ◦ γ .For any dense subcategory I v ∈ I defined by a class of maps v , this restricts11o a functor(1.10) γ ∗ : Fun v ( I, E ) → Fun γ ∗ v ( I ′ , E ) . If γ is essentially surjective, then (1.10) is faithful and conservative for any E . It is often useful to strengthen this in the following way. Definition 1.7.
A functor γ : I ′ → I between bounded categories I , I ′ isa localization if (1.10) is an equivalence for any v and E . A category I is simply connected if the tautological projection τ : I → pt is a localization. Example 1.8. If γ : I → I is a localization of small categories, then sois the opposite functor γ o : I o → I o (just replace E with E o in (1.10)). Inparticular, a small category I is simply connected if and only if so is I o . Example 1.9.
If a bounded full subcategory I ′ ⊂ I in a bounded category I is left or right-admissible in the sense of Definition 1.4, then the adjointfunctor γ † : I ′ → I is a localization in the sense of Definition 1.7. Moreover,any class of maps v in I ′ that contains the adjunction maps i → γ † ( γ ( i ))resp. γ † ( γ ( i )) → i for any i ∈ I ′ is dense in γ ∗† Iso .We of course assume known the notions of a limit and colimit, but to fixnotation, it is convenient to introduce the following.
Definition 1.10. An augmentation of a functor E : I → E between somecategories I , E is a functor E > : I > → E equipped with an isomorphism E > | I ∼ = E . A e -augmentation , for some e ∈ E , is an augmentation E > equipped with an isomorphism E > ( o ) ∼ = e . An augmentation E > is ex-act if for any other augmentation E ′ > , there exists a unique morphism ofaugmentations E > → E ′ > .Equivalently, an e -augmentation of a functor E : I → E is given by amorphism E → e I to the constant functor e I : I → E with value e , or by afactorization of E into a composition(1.11) I −−−−→ E /e ϕ −−−−→ E , where ϕ : E /e → E is the forgetful functor sending e ′ → e to e ′ . Bydefinition, E admits an exact augmentation E > if and only if the colimitcolim I E exists, and in this case, we have colim I E = E > ( o ). A category E is cocomplete if colim I F exists for any bounded I and F ∈ Fun(
I, F ).12 xample 1.11.
The category Sets is cocomplete. For any bounded cate-gory I , we denote π ( I ) = colim I pt , where pt : I → Sets is the constantfunctor that sends everything to a one-element set. Informally, π ( I ) isthe set of connected components of the category I , and I is connected if π ( I ) ∼ = pt . For any functor γ : I → I ′ between bounded cat-egories, and any category E , the left Kan extension γ ! E of a functor E ∈ Fun( I, E ) is a functor γ ! E ∈ Fun( I ′ , E ) equipped with a map a : E → γ ∗ γ ! E such that the composition map(1.12) Hom( γ ! E, E ′ ) γ ∗ −−−−→ Hom( γ ∗ γ ! E, γ ∗ E ) −◦ a −−−−→ Hom(
E, γ ∗ E ′ )is an isomorphism for any E ′ ∈ Fun( I ′ , E ). The left Kan extension is uniqueand functorial in E , if it exists. Example 1.12. If γ is a localization in the sense of Definition 1.7, then γ ! E exists for any E ∈
Fun γ ∗ v ( I, E ), for any class v , and γ ! provides the inverseequivalence to (1.10).If γ ! E exists for any E ∈ Fun( I, E ), then γ ! : Fun( I, E ) → Fun( I ′ , E )is the functor left-adjoint to γ ∗ . A degenerate example is the tautologicalprojection τ : I → pt : in this case, Fun( pt , E ) ∼ = E , providing a pair h e, a i of an object e ∈ E and a map a : E → τ ∗ e is equivalent to providinga e -augmentation E > : I > → E of the functor E , and h e, a i turns e intothe a Kan extension if and only if E > is exact, so that we simply have τ ! E = colim I E . In general, γ ! E is given by(1.13) γ ! E ( i ) = colim I/ γ i E | I/ γ i , E ∈ Fun( I, E ) , i ∈ I ′ , where I/ γ i are the left comma-fibers, and it exists iff so do the colimits inthe right-hand side.To check existence, it is often convenient to observe the following. Forany bounded category I and left-admissible subcategory I ′ ⊂ I , with theembedding functor γ : I ′ → I and its left-adjoint γ † : I → I ′ , the pullback γ ∗† is right-adjoint to the pullback γ ∗ , so that for any E ∈ Fun( I, E ), we have(1.14) colim I E ∼ = colim I ′ E | I ′ by adjunction, and the left-hand side exists iff so does the right-hand side.In many practical cases, this helps to compute left Kan extensions. Namely,for any functor γ : I → I ′ , a subcategory I ⊂ I ′ and an object i ∈ I ′ , say13hat an i -augmentation of the induced functor γ : I → I ′ is γ -admissible if the embedding I → I/ γ i of (1.11) is left-admissible, and define a framing of the functor γ as a collection of subcategories I ( i ) ⊂ I , one for each i ∈ I ′ , equipped with γ -admissible i -augmentations γ ( i ) > of the functors γ ( i ) = γ | I ( i ) . Remark 1.13. If γ : I → I ′ is fully faithful, then for any i ∈ I , I/ γ γ ( i ) ∼ = I/i has a terminal object. Thus in this case, to define a framing of γ , itsuffices to consider objects i ∈ I ′ that are not in γ ( I ) ⊂ I ′ , and then let I ( γ ( i )), i ∈ I be the point category pt embedded onto i ∈ I . Lemma 1.14.
Assume given bounded categories I , I ′ , a category E , and abounded functor γ : I → I ′ . Moreover, assume given a framing of γ . (i) For any E ∈ Fun( I, E ) , γ ! E exists if and only if for any i ∈ I , E | I ( i ) : I ( i ) → E admits an exact augmentation. (ii) For any E ∈ Fun( I ′ , E ) , E itself with the identity map γ ∗ E → γ ∗ E is aleft Kan extension γ ! γ ∗ E if and only if for any i ∈ I , the augmentation γ ( i ) ∗ > E of the functor γ ( i ) ∗ E is exact.Proof. Clear. (cid:3)
Example 1.15.
Say that a full embedding γ : I ′ → I is right-closed if I ( i ′ , i ) is empty for any i ′ ∈ I ′ and i ∈ I \ I ′ , and dually, say that γ is left-closed if γ o is right-closed (equivalently, γ is left resp. right-closed if I ′ ⊂ I isthe fiber I resp. I of a functor I → [1]). Then for a right-closed γ , all thecomma-fibers I ′ / γ i , i ∈ I \ I ′ in (1.13) are empty. Therefore by Remark 1.13,the left Kand extension γ ! E exists for any target category E with an initialobject o , and coincides with the canonical extension E < : I → E given by E < ( i ) = E ( i ) if i ∈ I ′ and o otherwise. A common example of a right-closedembedding is the embedding I ⊂ I < for any bounded I .Dually, the right Kan extension γ ∗ E is a pair h γ ∗ E, a i , γ ∗ E ∈ Fun( I ′ , E ), a : γ ∗ γ ∗ E → E satisfying the universal property dual to that of γ ! E , itis unique and functorial in E , and when it exists for any E , it providesa functor γ ∗ : Fun( I ′ , E ) → Fun( I, E ) right-adjoint to γ ∗ . In fact, if I and I ′ are small, so that there is no need to control boundedness, we have( γ ∗ ) o ∼ = γ ! E o , so that everything including Lemma 1.14 can be applied tothe right Kan extensions simply by passing to the opposite functors. Forlarge bounded categories, one can still compute right Kan extension γ ∗ by14sing an obvious dual version of (1.13) with limits and right comma-fibers,possibly combined with a framing of the opposite functor γ o , but one has tocheck that the result is a bounded functor. This is automatic in the situationof Example 1.12 (and in fact, in this case we have γ ! E ∼ = γ ∗ E ). This is alsoautomatic in Example 1.15, with appropriate dualization: γ : I ′ → I hasto be leftt-closed, o ∈ E must be the terminal object rather than the initialone, and we denote the canonical extension by E > .Let us now mention that quite often, the isomorphism (1.14) holds forfull embeddings that are not left-admissible. Here is one example. Definition 1.16.
A full subcategory I ′ ⊂ I is cofinal if for any i ∈ I , theright comma-fiber i \ I ′ is simply connected in the sense of Definition 2.10. Remark 1.17.
It is enough to check the condition of Definition 1.16 for i ∈ I \ i — indeed, i \ I ′ for i ∈ I ′ has an initial object id : i → i , thus it istautologically simply connected by Example 1.9.A left-admissible embedding γ : I ′ → I is automatically cofinal ( i \ I hasan initial object given by the adjunction map i → γ ( γ † ( i ))). The converseis not true. Nevertheless, for any cofinal full embedding γ : I ′ → I , the dualversion of (1.13) immediately shows that γ ∗ E exists for any constant functor E ∈ Fun( I ′ , E ), and the adjunction map E → γ ∗ γ ∗ E is an isomorphism forany constant functor E ∈ Fun( I, E ). Therefore for any E ∈ Fun( I, E ), westill have the isomorphism (1.14), and its source exists iff so does its target.Even more generally, γ ∗ E exists for a locally constant E , and γ ∗ induces anequivalence(1.15) γ ∗ : Fun ∀ ( I ′ , E ) ∼ = Fun ∀ ( I, E ) , so that I is simply connected if and only if so is I ′ . Lemma 1.18.
Assume given full embeddings I ′′ ⊂ I ′ ⊂ I of bounded cate-gories. Then I ′′ ⊂ I is cofinal if and only if I ′′ ⊂ I ′ and I ′ ⊂ I are cofinal.Proof. If I ′′ ⊂ I is cofinal, then I ′′ ⊂ I ′ is tautologically cofinal (the rightcomma-fiber i \ I ′′ for i ∈ I ′ does not change if we take i as an object of I ).But if I ′′ ⊂ I ′ is cofinal, then the embedding i \ I ′′ ⊂ i \ I ′ is cofinal for any i ∈ I — indeed, its right comma-fibers are the same as for I ′′ ⊂ I ′ — andwe are done by (1.15). (cid:3) emark 1.19. While a connected category I need not be simply connected,the pullback τ ∗ : E →
Fun( I, E ) with respect to the tautological projection τ : I → pt is still fully faithful for any target category E , so that colim I E = τ ! E and lim I E = τ ∗ E exist for any constant functor E : I → E , and wehave τ ! E ∼ = τ ∗ E , E ∼ = τ ∗ τ ! E ∼ = τ ∗ τ ∗ E . Thus if one only wants (1.14), onecan get away with a weaker notion of cofinality: it suffices to require thatthe right comma-fibers i \ I ′ are connected (as done in e.g. [KS, Chapter2.5]). However, we also need (1.15), so we use a stronger definition. In fact,all the cofinal embeddings in the paper will be also homotopy cofinal in thesense of Subsection 4.1 below. For any commutative ring k , we denote by k -mod the category of k -modules, and for any bounded category I , we sim-plify notation by writing Fun( I, k ) = Fun(
I, k -mod). This is an abeliancategory with enough projective and injectives, and we denote its derivedcategory by D ( I, k ). We shorten D ( pt , k ) to D ( k ). We have a natural functor(1.16) D : D ( I, k ) → Fun( I, D ( k )) , and for any dense subcategory in I defined by a class of morphisms v , wedenote by D v ( I, k ) ⊂ D ( I, k ) the full subcategory spanned by objects E with D ( E ) ∈ Fun v ( I, D ( k )). We recall that explicitly, objects in D ( I, k )are represented by chain complexes M q in the abelian category Fun( I, k ),and D ( I, k ) has a natural t -structure whose term D ≤ ( I, k ) ⊂ D ( I, k ) isspanned by complexes concentrated in non-negative homologocal degrees.For any bounded functor γ : I ′ → I , the pullback functor γ ∗ descendsto a functor γ ∗ : D ( I, k ) → D ( I ′ , k ) that has a left and a right-adjointfunctors L q γ ! , R q γ ∗ : D ( I ′ , k ) → D ( I, k ) obtained by taking the derivedfunctors of the left and right Kan extensions. For any i ∈ I , with theembedding ε ( i ) : pt → I onto i , the left Kan extension ε ( i ) ! is exact, andthe representable functor k i = ε ( i ) ! k ∈ Fun(
I, k ) is given by(1.17) k i ( i ′ ) = k [ I ( i, i ′ )] , i ′ ∈ I, where for any set S , k [ S ] is the free k -module generated by S . Sending i ∈ I to k i defines the fully faithful Yoneda embedding(1.18) Y : I o → Fun(
I, k ) . It may happen that the left Kan extension functor γ ! itself admits a left-ad-joint; here is a useful example of such a situation.16 xample 1.20. Assume given a full embedding γ : I ′ → I right-closed inthe sense of Example 1.15. Then γ ! is given by extension by 0, thus γ ! exact,and it in fact has a left-adjoint γ ! given by(1.19) γ ! = Y ! γ o ∗ Y ′ : Fun( I, k ) → Fun( I ′ , k ) , where Y , Y ′ are the Yoneda embeddings (1.18) for the categories I , I ′ , andthe right Kan extension γ o ∗ is again given by extension by 0. The derivedfunctor L q γ ! is then left-adjoint to γ ! ∼ = L q γ ! .For any functor M ∈ Fun(
I, k ), the homology of the category I withcoefficients in M is obtained by taking the total derived functor L q colim I M of the colimit functor colim I : we set(1.20) C q ( I, M ) = L q colim I M ∈ D ( k ) , and we denote by H q ( I, M ) the homology modules of the object (1.20). Both(1.14) and (1.13) have obvious counterparts for homology and derived leftKan extensions, and so does Lemma 1.14, so that one can compute derivedKan extensions by choosing a framing. For a useful generalization of (1.20),say that a k -valued bifunctor on I is a functor M ∈ Fun( I o × I, k ). Then forany map f : i → i ′ in I , we have a natural map(1.21) d f = M ( f o × id ) ⊕ ( − M ( id × f )) : M ( i ′ , i ) → M ( i, i ) ⊕ M ( i ′ , i ′ ) , and we can define the trace Tr I ( M ) of the bifunctor M by the exact sequence(1.22) L f ∈ I ( i,i ′ ) M ( i ′ , i ) d −−−−→ L i ∈ I −−−−→ Tr I ( M ) −−−−→ , where the sum on the left is over all maps f in I , and d is the sum of themaps (1.21) (Tr I M is also known as the “coend” of the functor M , see e.g.[Mc2, IX.6]). Then Tr I : Fun( I o × I, k ) → k -mod is right-exact, with thetotal derived functor L q Tr I , and we can define the bifunctor homology object CH q ( I, M ) ∈ D ( k ) by(1.23) CH q ( I, M ) = L q Tr I ( M ) . Bifunctor homology modules HH q ( I, M ) ∈ k -mod are then the homologymodules of the object (1.23). In particular, whenever we have two functors N ∈ Fun( I o , k ), M ∈ Fun(
I, k ), we can define their box product N ⊠ k M inFun( I o × I, k ) by N ⊠ k M ( i × i ′ ) = N ( i ) ⊗ k M ( i ′ ), and let(1.24) N ⊗ I M = Tr I ( N ⊠ k M ) . N , M gives the derivedtensor product N L ⊗ I M ∈ D ( k ), and as soon as either N or M is pointwise-flat — that is, takes values in the full subcategory k -mod fl ⊂ k -mod spannedby flat k -modules — we have a natural identification(1.25) N L ⊗ I M ∼ = CH q ( N ⊠ k M ) . Since any object in the derived category D ( I, k ) can be represented by acomplex of pointwise-flat k -modules, the product (1.25) extends to derivedcategories. The extended product admits a covariant version of the adjunc-tion isomorphism: for any functor γ : I ′ → I from some bounded I ′ , wehave a natural identification(1.26) E ⊗ I L q γ ! E ′ ∼ = γ o ∗ E ⊗ I ′ E ′ for any E ′ ∈ D ( I ′ , k ), E ∈ D ( I o , k ). In particular, for every M ∈ Fun(
I, k )and i ∈ I , we have a natural identification(1.27) k i L ⊗ I M ∼ = k i ⊗ I M ∼ = M ( i ) , where k i ∈ Fun( I o , k ) is the representable functor of (1.17) (that is automati-cally pointwise-flat). On the other hand, the constant functor k ∈ Fun( I o , k )with value k is also pointwise-flat, and we have(1.28) CH q ( I, π ∗ M ) = k L ⊗ I M ∼ = C q ( I, M ) , where π : I o × I → I is the projection. In this way, the functor homology(1.20) can be expressed as bifunctor homology (1.23). In the other direction,we have a canonical quasiisomorphism(1.29) CH q ( I, M ) ∼ = C q ( tw ( I ) , ( s × t ) ∗ M ) , where tw ( I ) is the twisted arrow category, and s × t : tw ( I ) → I o × I is theproduct of the projections (1.6) and (1.5). Example 1.21.
For a useful application of (1.28), assume given a boundedcategory I and a functor I → [1] with fibers I , I ⊂ I , as in Example 1.15,and define the homology of I with support in I with coefficients in some E ∈ D ( I, k ) as(1.30) C q ( I, I , E ) = C q ( I, L q j !1 E ) , H q ( I, I , E ) = H q ( I, L q j !1 E ) , j !1 is the functor (1.19) of Example 1.20 for the embedding j : I → I .Then we have(1.31) C q ( I, I , E ) ∼ = j o ∗ k L ⊗ I E, where k is the constant functor with value k , and j o ∗ is exact and given byextension by 0. Moreover, if we denote by j : I → I the other embedding,then j o is also exact and given by extension by 0, and we have a short exactsequence 0 −−−−→ j o k −−−−→ k −−−−→ j o ∗ k −−−−→ C q ( I , j ∗ E ) −−−−→ C q ( I, E ) −−−−→ C q ( I, I , E ) −−−−→ of homology complexes and the corresponding long exact sequence of homol-ogy groups. For example, for any bounded category I , we have the projection I > → [1] sending I ⊂ I > to 0 and the new terminal object o ∈ I > to 1; thenfor any augmented functor E ∈ Fun( I > , k ), we have C q ( I > , E ) ∼ = E ( o ), thefirst map in (1.32) is the augmentation map, and it is a quasiisomorphismif and only if H q ( I > , { o } , E ) = 0.We say that a category I is k -linear if it is enriched over k -mod — thatis, all the sets I ( i, i ′ ) are equipped with a k -module structure so that thecomposition maps are bilinear. We say that a k -linear category I is flat if so are all the k -modules I ( i, i ′ ). For any k -linear bounded category I ,we denote by Fun k ( I, k ) ⊂ Fun(
I, k ) be the full subcategory spanned by k -linear functors, and we let D ( I ) be its derived category. More generally,a small DG category I q is a small category I enriched over the category C q ( k -mod fl ) of chain complexes of flat k -modules — that is, we are given acomplex I q ( i, i ′ ) for any i, i ∈ I ′ with I ( i, i ′ ) = I ( i, i ′ ) as sets, and the unitaland associative k -linear composition maps I q ( i, i ′ ) ⊗ k I q ( i ′ , i ′′ ) → I q ( i, i ′′ ) thatextend the compositions in I . We assume known that every DG category I q has a derived category D ( I q ) of DG modules with the standard propertiesthat can be found for example in [Ke].For any k -linear category I , we let Fun k ( I o × I, k ) ⊂ Fun( I o × I, k )be the full subcategory spanned by bifunctors that are k -linear in each ofthe two variables, and we let Tr kI be the restriction of the functor (1.22)to Fun k ( I o × I, k ). For any M ∈ Fun k ( I o × I, k ), the k -linear bifunctorhomology object of I with coefficients in M is defined by CH q ( I/k, M ) = L q Tr kI ( M ) ∈ D ( k ) , k -linear bifunctor homology modules HH q ( I/k, M ) ∈ k -mod are itshomology modules. The embedding Fun k ( I o × I, k ) ⊂ Fun( I o × I, k ) theninduces a functorial map(1.33) CH q ( I, M ) → CH q ( I/k, M )for any M ∈ Fun k ( I o × I, k ). If I is flat, we also have the obvious k -linearversion − ⊗ I/k − of the product (1.24), with its derived version − L ⊗ I/k − ,and the identification (1.25) if M or N is pointwise-flat. In particular, I ( − , i ) : I o → k -mod is then k -linear and pointwise-flat, and we also havethe k -linear version of the Yoneda isomorphism (1.27). Example 1.22. A k -linear category I with one object is the same thing asan associative unital k -algebra A , and it is flat iff so is A . The categoryFun k ( I o × I, k ) is then the category A -bimod of A -bimodules — that is, leftmodules over A o ⊗ k A — and Tr I ( M ) coincides with Tr A ( M ) = M/ [ A, M ],where [
A, M ] ⊂ M is the k -submodule spanned by commutators am − ma , a ∈ A , m ∈ M . The k -linear Yoneda isomorphism (1.27) reads as(1.34) L q Tr A ( A o ⊗ k M ) ∼ = Tr A ( A o ⊗ k M ) ∼ = M, for any flat k -algebra A and left A -module M . We assume known the machinery ofGrothendieck fibrations and cofibrations of [Gr]. As a reminder, a functor γ : C → I is a precofibration if for any i ∈ I , the tautological embedding C i ⊂ C /i admits a left-adjoint functor(2.1) ζ ( i ) : C /i → C i . In this case, for any morphism f : i ′ → i in the category I , we have anembedding C i ′ → C /i , c
7→ h c, f i , and composing it with ζ ( i ) gives a functor f ! : C i ′ → C i known as the transition functor of the precofibration γ . Amorphism g : c ′ → c in C is cocartesian over I if ζ ( γ ( c )) inverts its naturallifting to a morphism in C /c . For a composable pair of maps f , f ′ , theadjunction provides a natural map(2.2) ( f ◦ f ′ ) ! → f ! ◦ f ′ ! , and a precofibration is a cofibration if all these maps are isomorphisms.20 xample 2.1. The cylinder C ( γ ) of a functor γ : C → C is the categorywhose objects are those of C and C , and whose morphisms are given by(2.3) C ( γ )( c, c ′ ) = C i ( c, c ′ ) , c, c ′ ∈ C i , i = 0 , C ( γ ( c ) , c ′ ) , c ∈ C , c ′ ∈ C ∅ otherwise . If τ : I → pt is the tautological projection from some category I to thepoint, then C ( τ ) ∼ = I > , and if id : pt → pt is the identity functor from thepoint category to itself, then C ( id ) ∼ = pt > ∼ = [1]. In general, we have theprojection χ : C ( γ ) → [1] with fibers C , C , and it is a cofibration withtransition functor γ . Conversely, for any cofibration C → [1] with transitionfunctor γ : C → C , we have C ∼ = C ( γ ). Example 2.2.
Assume given a cofibration π : C → I . Then its canonicalextension π > : C > → I > is also a cofibration; its fiber C >o over the newterminal object o ∈ I > consists of the terminal object o ∈ C > , and thetransition functor C >i ∼ = C i → C >o = pt corresponding to any i ∈ I is thetautological projection.More generally, extending π to a precofibration π ′ : C ′ → I > is equivalentto giving a category E = C ′ o and a functor γ = ζ ( o ) : C → E ; for C > , onetakes pt as E and the tautological projection C → pt as γ .A functor C → I is a fibration if the opposite functor C o → I o is acofibration. Thus a cofibration C → I defines the opposite fibration C o → I o .It also defines the transpose fibration C ⊥ → I o that has the same fibers C ⊥ i = C i , i ∈ I , the same transition functors f ∗ = f ! : C i → C i ′ for any map f : i → i ′ in I , and the isomorphisms (2.2) inverse to the same isomorphismsfor C . Dually, a fibration C → I defines the opposite cofibration C o → I o andthe transpose cofibration C ⊥ → I o . Moreover, a fibration C → I can also bea cofibration; this happens if all the transition functors f ∗ have left-adjointfunctors f ! (and these are then the cofibration transition functors). In sucha case, one says that C → I is a bifibration . Example 2.3.
For any category I , the projections (1.5) are cofibrations,projection ar ( I ) → I of (1.6) is a fibration, and the projection tw ( I ) → I o is its transpose cofibration. Example 2.4.
For any functor γ : I ′ → I , and any cofibration, resp. fibra-tion, resp. bifibration C → I , the pullback γ ∗ C → I ′ is a cofibration resp.fibration resp. bifibration. 21 xample 2.5. A full embedding I ′ → I is a fibration resp. cofibration ifand only if it is left resp. right-closed in the sense of Example 1.15.For any category I and categories C , C ′ equipped with functors π : C → I , π ′ : C ′ → I , a functor from C to C ′ over I is a pair of a functor γ : C → C ′ and an isomorphism α ( γ ) : π ′ ◦ γ ∼ = π . If γ has a left-adjoint γ † : C → C ′ ,then we have a map α † ( γ ) : π ′ → π ◦ γ † adjoint to α ( γ ), and we say that γ † is a left-adjoint over I if α † ( γ ) is an isomorphism. Dually, a right-adjoint γ † is a right-adjoint over I if α † ( γ ) : π ◦ γ † → π ′ is an isomorphism.If C is bounded, and we are given another category ϕ : E → I over I , thenbounded functors from C to E over I form a well-defined category Fun I ( C , E ).If I itself is bounded, we can let C = I ; then Sec( I, E ) = Fun I ( I, E ) isthe category of sections I → E of the projection ϕ . If we have a functor γ : C ′ → C over I between bounded categories π : C → I , π ′ : C ′ → I , thenjust as in the absolute case, we have the pullback functor γ ∗ : Fun I ( C ′ , E ) → Fun I ( C , E ). For any E ∈ Fun I ( C , E ), the left Kan extension γ I ! E over I isa pair of a functor γ I ! E ∈ Fun I ( C ′ , E ) and a map a : E → γ ∗ γ I ! E such that(1.12) is an isomorphism for any E ′ ∈ Fun I ( C ′ , E ). Just as in the absolutecase, γ I ! E is unique and functorial, if it exists.If ϕ : E → I is a cofibration, then γ I ! E can be computed by a relativeversion of (1.13). Namely, for any c ∈ C ′ , the functor E defines a functor E | C / γ c : C / γ c → E / ϕ π ′ ( c ) sending h c ′ , α ( c ′ ) i ∈ C / γ c to E ( c ′ ) equipped withthe composition map ϕ ( E ( c ′ )) α ( E )( c ′ ) −−−−−→ π ( c ′ ) α ( γ )( c ′ ) − −−−−−−→ π ′ ( γ ( c ′ )) π ′ ( α ( c ′ )) −−−−−→ π ′ ( c ) , and we then have(2.4) γ I ! E ( c ) = colim C / γ c ζ ( π ′ ( c )) ◦ E | C / γ c , E ∈ Fun I ( C , E ) , c ∈ C ′ , where ζ ( − ) is the functor (2.1) for the cofibration ϕ : E → I . Moreover γ I ! E exists iff so do the colimits in the right-hand side of (2.4). We also havean obvious relative counterpart of Lemma 1.14: for any framing {C ( c ) ⊂C , γ ( c ) > : C ( c ) > → C ′ } , c ∈ C ′ of the functor γ : C → C ′ , we have(2.5) colim C/ γ c ζ ( π ′ ( c )) ◦ E | C/ γ c ∼ = colim C ( c ) ζ ( π ′ ( c )) ◦ E | C ( c ) for any E ∈ Fun( C , E ), c ∈ C ′ , where ζ ( π ′ ( c )) acts on E | C ( c ) through thefactorization (1.11) of the augmentation γ ( c ) > , and for any E ′ ∈ Fun I ( C ′ , E ),we have E ′ ∼ = γ I ! γ ∗ E ′ iff for any c ∈ C ′ , the augmented functor ζ ( π ′ ( c )) ◦ γ ( c ) ∗ > E is exact. 22 xample 2.6. If γ : C → C ′ admits a right-adjoint γ † : C ′ → C ′ , a framingof γ is obtained by taking C ( c ) = pt embedded onto γ † ( c ) ∈ C and augmentedby the adjunction map a : γ ( γ † ( c )) → c . Then (2.4) and (2.5) show that γ I ! E exists for any cofibration ϕ : E → I and functor E : C → E over I , andit is given by(2.6) γ I ! E ( c ) = π ( a ) ! E ( γ † ( c )) , where π ( a ) ! is the transition functor for the cofibration C associated to themap π ( a ) : π ( γ † ( c )) ∼ = π ′ ( γ ( γ † ( c ))) → π ′ ( c ).A precofibration γ : C → I is bounded resp. discrete if so are all its fibers C i , and strongly discrete if γ ∗ Id = Id . A discrete precofibration is of coursetautologically a cofibration. A functor X : I → Sets defines a boundedstrongly discrete cofibration I [ X ] → I , where I [ X ] is the category of pairs h i, x i , i ∈ I , x ∈ X ( i ), with maps h i, x i → h i ′ , x ′ i given by maps f : i → i ′ such that f ( x ) = x ′ , and the projection I [ X ] → I sending h i, x i to i . Everybounded strongly discrete cofibration is of this form. More generally, forany bounded confibration γ : C → I , we denote by π ( C /I ) → I the discretecofibration corresponding to the functor γ ! pt : I → Sets. Then we have anatural functor γ : C → π ( C /I ), the functor is an equivalence if and only if C/I is discrete, and for any functor ϕ : C → C ′ over I to a bounded discretecofibration C ′ → I with the corresponding equivalence γ ′ : C ′ → π ( C ′ /I ),the composition γ ′ ◦ ϕ factors uniquely through γ . Example 2.7.
For any category I with the twisted arrow category tw ( I ),sending f : i ′ → i to i ′ × i gives a discrete cofibration tw ( I ) → I o × I corresponding to the Hom-functor I ( − , − ) : I o × I → Sets; projecting furtherdown to I o , we get the cofibration (1.6).We will say that a cofibration γ : C → I is semidiscrete if γ ∗ Iso = Iso , orequivalently, all its fibers C i are groupoids, or equivalently, all maps in C arecocartesian over I . For any cofibration C → I , we have a dense subcategory C ♮ ⊂ I defined by the class ♮ of cocartesian maps, the induced projection C ♮ → I is a semidiscrete cofibration, and for any semidiscrete cofibration C ′ → I , any functor C ′ → C cocartesian over I factors uniquely through C ♮ .For any semidiscrete cofibration γ : C ′ → I and any functor π : C ′ → C such that γ ◦ π : C ′ → I is a cofibration, π itself is a cofibration. A discretecofibration is semidiscrete (although the opposite is not true). Dually, afibration C → I is semidiscrete if so is the opposite cofibration C o → I o ,and for any fibration C → I , we have the maximal semidiscrete subfibration23 ‡ ⊂ C defined by the class ‡ of all cartesian maps. Note that if a cofibration C → I is semidiscrete, then so is the transpose fibration C ⊥ → I o , and wein fact have C ⊥ ∼ = C o . If both π : C → I and π ′ : C ′ → I are cofibra-tions, then a functor γ : C → C ′ over I is explicitly given by a collection offunctors γ i : C i → C ′ i between their fibers, and maps(2.7) γ f : f ′ ! ◦ γ i → γ i ′ ◦ f ! , one for each morphism f : i → i ′ in I , subject to compatibility conditions(where f ! , f ′ ! are transition functors of the cofibrations π , π ′ ). The functor γ is cocartesian over f iff (2.7) is an isomorphism, and it is cocartesian if it iscocartesian over all maps in I . The terminology for fibrations is dual (with“cocartesian” replaced by “cartesian”). A functor is cocartesian if and onlyif it sends all cocartesian maps to cocartesian maps. If C and I are bounded,then for any class of maps v in I , we denote by Fun vI ( C , C ′ ) ⊂ Fun I ( C , C ′ )the full subcategory spanned by functors cocartesian over all maps in v . Welet Sec v ( I, C ) = Fun vI ( I, C ) be the category of sections cocartesian over mapsin v , and dually, for any fibration C → I , we let Sec v ( I, C ) ⊂ Sec( I, C ) bethe category of section cartesian over maps in v . We note that speaking offunctors and sections cartesian resp. cocartesian over v makes sense even if C → I is not a fibration resp. cofibration — it suffices to assume that it issuch over I v ⊂ I . For any two cofibration C , C → I , a functor γ : C → C cocartesian over I induces a functor γ ⊥ : C ⊥ → C ⊥ between the transposefibrations cartesian over I , and vice versa (with γ i = γ ⊥ i , i ∈ I , and the maps(2.7) for γ ⊥ inverse to those for γ ). For any cofibration C → I , sending E to E ⊥ provides an equivalence(2.8) Sec ∀ ( I, C ) ∼ = Sec ∀ ( I o , C ⊥ ) . A convenient framing for a cocartesian functor γ : C → C ′ over I is given bythe left comma-fibers of the functors γ i : one takes C ( c ) = C π ′ ( c ) / γ π ′ ( c ) c , withthe obvious augmentations. In particular, if C ′ = I , π ′ = id , then γ ∼ = π isautomatically cocartesian, and has a framing(2.9) C ( i ) = C i C ′ = ϕ ∗ C ρ −−−−→ C γ ′ y y γ I ′ ϕ −−−−→ I and γ is a cofibration, then for any functor E ∈ Fun( C , E ) to some category C , γ ! E exists iff so does γ ′ ! ρ ∗ E , and the adjunction map(2.11) γ ′ ! ρ ∗ E → ϕ ∗ γ ! E is an isomorphism. We call it the base change isomorphism associated to asquare (2.10). Example 2.8.
Assume given a bounded discrete cofibration γ : I ′ → I ,and a functor X ′ : I ′ → Sets. Then by (2.11), X = γ ! X ′ is given by(2.12) X ( i ) ∼ = a i ′ ∈ I i X ′ ( i ′ ) , i ∈ I, and this implies that I ′ X ′ ∼ = IX (with the discrete cofibration IX → I obtained by composing γ with I ′ X ′ → I ′ ). Example 2.9.
As another application of (2.11), assume given a small cat-egory I and a category E , and note that we have an obvious equivalenceFun ∀ ( I, E ) ∼ = Fun ∀ ( I o , E ). Then the same equivalence can be realized byKan extensions. Namely, consider the twisted arrow category tw ( I o ), withthe projections s : tw ( I o ) → I , t : tw ( I o ) → I o of (1.5), (1.6), and for any E ∈ Fun( I, E ), let(2.13) Tw I ( E ) = t ! s ∗ E whenever the Kan extension exist. Then t is a cofibration with fibers tw ( I o ) i ∼ = i \ I , i ∈ I o , and for any E ∈ Fun( I, E ), we have s ∗ E | tw ( I o ) i ∼ = p ∗ i E ,where p i is the projection (1.1). Since i \ I has an initial object, this impliesthat as soon as E is locally constant, we haveTw I ( E )( i ) ∼ = colim i \ I p ∗ i E ∼ = E ( i ) , so that Tw I ( E ) exists. Moreover, for any morphism f : i → i ′ , the morphismTw I ( E )( f ) is inverse to E ( f ), so that Tw I ( E ) is locally constant. Thus Tw I gives the desired equivalence. 25 emma 2.10. Assume given two bounded cofibrations C , C ′ → I and a co-cartesian functor γ : C → C ′ over I . Moreover, assume that for any i ∈ I ,the induced functor γ i : C i → C ′ i between the fibers (i) is full, or (ii) has aright-adjoint γ † i , or (iii) is opposite to a cofinal full embedding, or (iv) is alocalization in the sense of Definition 1.7. Then the same holds for γ .Proof. (i) is obvious. For (ii), γ † has components γ † i , with the maps (2.7)adjoint to those for γ . For (iii), the comma-fibers of γ i are left-admissiblein the comma-fibers of γ , so the claim follows from (1.15). Finally, for(iv), it suffices to check that the left Kan extension γ ! E exists for any E ∈ Fun γ ∗ Iso ( C , E ), the adjunction map E → γ ∗ γ ! E is an isomorphism, and soid the adjunction map γ ! γ ∗ E ′ → E ′ for any E ′ ∈ Fun( C, E ). By (2.11), allof this can be checked after restricting to the fibers over all i ∈ I . (cid:3) Lemma 2.11.
Let
C → I be a cofibration, and C ′ ⊂ C a full subcategory. (i) Assume that for any morphism f : i → i ′ in I , the transition functor f ! : C i → C i ′ sends C ′ i ⊂ C i into C ′ i ′ ⊂ C i ′ . Then C ′ → I is a cofibration,and the embedding C ′ ⊂ C is cocartesian. (ii) Assume that for any i ∈ I , the embedding C ′ i ⊂ C i has a left-adjoint λ i : C i → C ′ i . Moreover, say that a map g in C i , i ∈ I is λ -trivial if λ i ( g ) is invertible, and assume that for any map f in I , f ! sends λ -trivial maps to λ -trivial maps. Then the embedding C ′ ⊂ C admits aleft-adjoint λ : C → I over I . (iii) Assume that the embedding C ′ → C admits a left-adjoint λ : C → C ′ over I . Then C ′ → I is a cofibration, and λ is cocartesian over I .Proof. The first claim is obvious. For the second and the thrid, the transitionfunctors for C ′ are f ′ ! = λ i ′ ◦ f ! for any f : i → i ′ in I , and functor λ in (iii)is given by λ i with the maps (2.7) given by adjunction. This shows that C ′ → I is a precofibration. Then either the fact that λ is adjoint to theembedding, or the final condition in (iii) is sufficient to conclude the maps(2.2) and (2.7) are isomorphisms. (cid:3) Example 2.12.
Assume given a category equipped with a factorizationsystem h L, R i in the sense of Definition 1.1, and consider the subcategory ar R ( I ) ⊂ ar ( I ) and the cofibration (1.5). Then by Example 1.6, the fibers ar R ( I ) i ⊂ ar ( I ) i are left-admissible subcategories, with the adjoint functors λ i : ar ( I ) i → ar R ( I ) i sending an arrow f : i ′ → i to the component r : i ′′ → i
26f its decomposition (1.8), and a map g is λ -trivial iff s ( g ) ∈ L . Since thetransition functors for the cofibration (1.5) commute with s , we are in thesituation of Lemma 2.11 (ii), so that the induced projection t : ar R ( I ) → I is a cofibration. Dually, s : ar L ( I ) → I induced by (1.6) is a fibration.By abuse of terminology, for any cofibration π : C → I , we will say thata functor γ : E → C from some category E is cocartesian over I if γ ( f ) iscocartesian for any map f in E , and vertical over I if π ◦ γ is constant. If E is bounded, then bounded vertical functors form a cofibration Fun( E , C /I )over I with fibers Fun( E , C /I ) i ∼ = Fun( E , C i ), i ∈ I . We will also say that afunctor γ : C → E is cocartesian over I if it inverts all maps in C cocartesianover I (or equivalently, if γ × id : C → E × I is cocartesian over I , orequivalently, if the corresponding precofibration C ′ → I > of Example 2.2 isa cofibration). For fibrations, the terminology is again dual.As a complement to Example 2.12, note that for any cofibration π : C → I , any map f : c → c ′ in C has a factorization(2.14) c c −−−−→ c ′′ v −−−−→ c ′ with cocartesian c and v ∈ π ∗ Iso , and (2.14) is unique up to a uniqueisomorphism, so that h ♮, π ∗ Iso i is a factorization system on C . We have ar ♮ ( C ) ∼ = C × sI ar ( I ), where s is the fibration (1.6). Example 2.13.
In the situation of Example 2.12, maps in ar R ( I ) cocarte-sian with respect to t : ar R ( I ) → I correspond to diagrams (1.3) with g ′ ∈ L .Then s : ar R ( I ) → I restricts to a projection(2.15) s : ar R ( I ) ♮ → I L , and the dense embedding β : I L → I factors as(2.16) I L η −−−−→ ar R ( I ) ♮ t −−−−→ I, where η is induced by (1.7) and left-adjoint to the projection (2.15). Inparticular, we have β ! ∼ = t ! ◦ η ! ∼ = t ! ◦ s ∗ , whenever these Kan extensionsexist. For any cofibration
C → I , a functor E : C → E cocartesian over I defines a cocartesian functor ( E × id ) : C → E × I over I , thus a transpose functor ( E × id ) ⊥ : C ⊥ → ( E × I ) ⊥ ∼ = E × I o , and then( E × id ) ⊥ = E ⊥ × id for a unique functor E ⊥ : C ⊥ → E cartesian over I o . If27 is small and C is bounded, then C ⊥ is bounded, and the correspondence E E ⊥ provides an equivalence(2.17) Fun ♮ ( C , E ) ∼ = Fun ‡ ( C ⊥ , E ) , where Fun ♮ ( C , E ) ⊂ Fun( C , E ) resp. Fun ‡ ( C ⊥ , E ) ⊂ Fun( C ⊥ , E ) are the fullsubcategories spanned by cocartesian resp. cartesian functors. Alternatively,as in Example 2.9, (2.17) can be realized by Kan extensions. Namely, con-sider the twisted arrow category tw ( I ) of Example 2.3, and let tw ( C /I ) = tw ( I ) × I C , tw ⊥ ( C /I ) = C ⊥ × I o tw ( I ). For any f : i ′ → i considered as anobject in tw ( I ), we have the transition functor f ! : C i ′ → C i , and taken to-gether, these functors provide a functor q : tw ⊥ ( C /I ) → tw ( C /I ) over tw ( I ).We then have the diagram(2.18) C r ←−−−− tw ( C /I ) q ←−−−− tw ⊥ ( C /I ) l −−−−→ C ⊥ , where l and r are the projection functors, and for any cocartesian E : C → E ,we have E ⊥ ∼ = l ! q ∗ r ∗ E .For any cofibration C → I over a category I with an initial object o ∈ I ,the embedding C o → C uniquely extends to a functor(2.19) σ : C o × I → C cocartesian over I (explicitly, σ sends c × i to f ( i ) ! c , where f : o → I is theunique map). This observation has the following useful generalization. Let ϕ : I ′ → I be a bounded functor between bounded categories. Then for anycofibration C → I ′ , we can define a cofibration ϕ ∗∗ C → I with fibers(2.20) ϕ ∗∗ C i = Sec( i \ ϕ I ′ , p ∗ i C ) , i ∈ I, where p i is the projection (1.2), and transition functors f ! = ( f ∗ ) ∗ , where f ∗ is the functor (1.2). For any class v of maps in I ′ , we let(2.21) ϕ v ∗∗ C ⊂ ϕ ∗∗ C be the subcofibration spanned by sections cocartesian over maps in p ∗ i v , i ∈ I , and we let ϕ ∗ C = ϕ ∀∗∗ C be the subcofibration spanned by cocartesiansections. Then since for any i ∈ I , the right comma-fiber i \ I has the initialobject, (2.19) induces a canonical equivalence id ∗ C ∼ = C for any cofibration C → I , and then the pullback functors p ∗ i , i ∈ I provide a functor(2.22) C → ϕ ∗ ϕ ∗ C I . On the other hand, for any cofibration C → I ′ , theevaluation functors (1.9) provide a functor(2.23) ev : ϕ ∗ ϕ ∗∗ C → C over I ′ whose restriction to ϕ ∗ ϕ ∗ C ⊂ ϕ ∗ ϕ ∗∗ C is cocartesian over I ′ . By virtueof the universal property of the functor (1.9), for any bounded cofibration C ′ → I and any cofibration C → I ′ , the functors (2.22) and (2.23) providean equivalence(2.24) Fun I ′ ( ϕ ∗ C ′ , C ) ∼ = Fun ♮I ( C ′ , ϕ ∗∗ C ) , where Fun ♮I ( − , − ) ⊂ Fun I ( − , − ) is the full subcategory spanned by co-cartesian functors. Moreover, for any class of maps v in I ′ , a functor E : ϕ ∗ C ′ → C is cocartesian over maps in v if and only if the correspondingfunctor E ′ : C ′ → ϕ ∗∗ C factors through ϕ v ∗∗ C ⊂ ϕ ∗∗ C , and in particular,(2.24) induces an equivalence(2.25) Fun ∀ I ′ ( ϕ ∗ C ′ , C ) ∼ = Fun ∀ I ( C ′ , ϕ ∗ C ) , a sort of an adjunction property for ϕ ∗ and ϕ ∗ . By adjunction, for a com-posable pair ϕ : I ′ → I , ϕ ′ : I ′′ → I ′ of bounded functors, we have anatural equivalence ϕ ∗ ϕ ′∗ C ∼ = ( ϕ ◦ ϕ ′ ) ∗ C for any cofibration C → I ′′ , and if ϕ : I ′ → I has a left-adjoint ϕ † : I → I ′ , then we have a natural equiv-alence ϕ ∗ C ∼ = ϕ ∗† C for any cofibration C → I ′ . In particular, if we have aright-admissible full subcategory I ′ ⊂ I in some bounded I , with the em-bedding functor γ : I ′ → I , and a cofibration C → I , then (2.25) providesan equivalence(2.26) Sec ∀ ( I, C ) ∼ = Sec ∀ ( I ′ , γ ∗ C ) . This implies that just as for usual Kan extensions, one can compute the fibersof ϕ ∗ C by choosing a framing I ′ ( i ) for the opposite functor ϕ o in the sense ofLemma 1.14, and replacing the fiber categories i \ ϕ I ′ in (2.20) with I ′ ( i ). If ϕ is a fibration, one can use the framing (2.9). Moreover, (2.26) is actuallyinduced by the pullback functor γ ∗ , with the inverse equivalence providedby the pullback γ ∗† with respect to the right-adjoint functor γ † : I → I ′ , andthen more generally, for any class v of maps in I ′ , the pullbacks γ ∗ and γ ∗† provide an equivalence(2.27) Sec γ ∗† v ( I, C ) ∼ = Sec v ( I ′ , γ ∗ C ) . In good situations, this allows to use framings to compute ϕ v ∗∗ for moregeneral classes v . 29 xample 2.14. Assume given a bounded category I equipped with a fac-torization system h L, R i in the sense of Definition 1.1, and consider theprojections s, t : ar L ( I ) → I of Example 2.12. Then for any cofibration C → I , the pullback functor t ∗ induces a functor(2.28) Id R ∗∗ C → s ‡∗∗ t ∗ C over I , where ‡ = t ∗ ( R ) is the class of maps in ar L ( I ) cartesian over I , and(2.27) for the admissible subcategories of Example 1.6 immediately showsthat (2.28) is an equivalence. Example 2.15.
For any bounded cofibration π : C → I over a boundedcategory I , and another cofibration C ′ /I , let(2.29) Fun ∀ ( C /I, C ′ /I ) = π ♮ ∗∗ π ∗ C ′ , where ♮ is the class of maps cocartesian over I . Then Fun ∀ ( C /I, C ′ /I ) → I is a cofibration with fibersFun ∀ ( C /I, C ′ /I ) i ∼ = Fun p ∗ i ♮i \ I ( p ∗ i C , p ∗ i C ′ ) , i ∈ I, where p i is the projection (1.1), and (2.25) provides an equivalence(2.30) Fun ∀ I ( C , C ′ ) ∼ = Sec ∀ ( I, Fun ∀ ( C /I, C ′ /I )) . Informally, the category of categories cofibered over I and cocartesian func-tors between them is cartesian-closed, with (2.29) as the mapping category.Slightly more generally, for any bounded precofibration π : C → I , andfor any category E , we can define a precofibration Fun( C /I, E ) → I o withfibers Fun( C /I, E ) i = Fun( C i , E ), transition functors f o ! = ( f ! ) ∗ induced bythe transition functors f ! of the cofibration π , and the maps (2.2) inducedby the corresponding maps for π . If π is a cofibration, then Fun( C /I, E ) isalso a cofibration, and we in fact have(2.31) Fun( C /I, E ) ∼ = π ⊥‡∗∗ τ ∗ E , where π ⊥ : C ⊥ → I o is transpose to π , τ : C ⊥ → pt is the tautological pro-jection, and ‡ is the class of maps cartesian over I o , as in Example 2.14. Inthis case, (2.24) identifies functors from C to E and sections of the transposefibration Fun( C /I, E ) ⊥ → I , and this correspondence identifies cocartesianfunctors with cartesian sections. By (2.17), these in turn correspond tococartesian sections of the cofibration Fun( C /I, E ) → I o .30 .4 Kernels and reflections. In practical applications, it is useful tocombine the pullback and pushforward operations on cofibrations, as in Ex-ample 2.14. Here is one specific construction of this type that we will need.
Definition 2.16.
For any bounded category I , an I -kernel is a category K equipped with a bounded cofibration K → I o × I . A morphism between I -kernels K , K is a functor γ : K → K ′ cocartesian over I o × I .For any bounded category I with an I -kernel K , the structural cofibration K → I o × I is the product s × t of functors s : K → I o , K : I ′ → I . Both arecofibrations, and each of them is cocartesian with respect to the other one.We can then consider the fibration T = t ⊥ : K ⊥ → I o transpose to t , andbeing cocartesian, s induces a cartesian functor S : K ⊥ → I o that is again acofibration. Then for any cofibration C → I o , we let(2.32) K ⊗ I C = T ♮ ∗∗ S ∗ C , where ♮ is the class of maps in K ⊥ cocartesian with respect to S . This is acofibration over I o with fibers(2.33) ( K ⊗ I C ) i ∼ = Fun ♮I o ( K i , C ) , i ∈ I, where K i stands for the fiber of the cofibration t cofibered over I o by thecofibration s . For any morphism γ : K → K ′ of I -kernels, the evaluationfunctor (2.23) for ϕ = γ ⊥ then induces a functor(2.34) K ′ ⊗ I C → K ⊗ I C , cartesian over I o . In terms of (2.33), its component over some i ∈ I is thepullback γ ∗ i with respect to the functor γ i : K i → K ′ i . Example 2.17.
For any bounded category I , the twisted arrow category tw ( I ) with its discrete cofibration s × t : tw ( I ) → I o × I of Example 2.7 is an I -kernel. The category tw ( I ) ⊥ is opposite to the arrow category ar ( I ) o , withthe projections S = s o , T = t o opposite to s of (1.6) and t of (1.5). Then(1.7) provides their common section η : I → ar ( I ), and for any cofibration C → I o , we have a natural equivalence tw ( I ) ⊗ I C ∼ = T ∗ S ∗ C ∼ = T ∗ η ∗ C ∼ = C , where since s : tw ( I ) i → I o is discrete for any i ∈ I , we can replace T ♮ ∗∗ in(2.32) with T ∗ . 31 xample 2.18. For any two cofibrations C , C ′ → I with C bounded, let π : C ⊥ → I o be the transpose fibration, and let(2.35) Fun I ( C , C ′ ) = ( π × id ) ‡∗∗ ( π × id ) ∗ t ∗ C ′ , where t : I o × I → I is the projection, and ‡ is the class of maps in C ′ × I cartesian over I o × I . Then (2.35) is an I -kernel, and for any class of maps v in I , we have a natural identification(2.36) Fun t ∗ vI o × I ( tw ( I ) , Fun I ( C , C ′ )) ∼ = Fun vI ( C , C ′ ) . In particular, morphisms of I -kernels tw ( I ) → Fun I ( C , C ′ ) correspond tococartesian functors C → C ′ .In favourable circumstances, one can use Example 2.18 to establish aversion of (2.30) for functors that are only cocartesian over a class of maps.To do this, we first observe that the transpose fibration construction worksin families. Namely, for any cofibration J → I , denote by J ⊳ = J ⊥ o → I thecofibration with fibers J oi and transition functors opposite to those of J → I ,and note that a cocartesian functor γ : J → J between two cofibrations J , J /I gives rise to a cocartesian functor γ ⊳ : J ⊳ → J ⊳ . Then for any pairof cofibrations J → I , γ : C → J , we can define the reflection ( C /J ) ⊳ by(2.37) ( C/J ) ⊳ = ( C ⊥ ) ⊥ , where the transpose fibration C ⊥ is taken over J , and then the transposecofibration is taken over I o . We have ( J/J ) ⊳ = J ⊳ , and in general, ( C/J ) ⊳ is equipped with a cofibration ( C/J ) ⊳ → I and a functor γ ⊳ : ( C/J ) ⊳ → J ⊳ cocartesian over I . Over each i ∈ I , the fiber γ ⊳ i : ( C /J ) ⊳ i → J ⊳ i ∼ = J oi isthe fibration transpose to γ i . If we let v , c be classes of maps in J and J ⊳ vertical resp. cocartesian over I , then γ ⊳ is a cofibration over J ⊳ c and afibration over J ⊳ v , and we have an equivalence(2.38) Sec v ( J, C ) ∼ = Sec v ( J ⊳ , ( C /J ) ⊳ ) , a relative version of (2.8). For any commutative square(2.39) j c −−−−→ j v y y v j c −−−−→ j in J ⊳ with c , c ∈ c , v , v ∈ v , we have an isomorphism c ◦ v ∗ ∼ = v ∗ ◦ c .32 emma 2.19. Assume given a cofibration J → I and a cofibration γ : C → J such that for any map v : j → j ′ in J ⊳ vertical over I , the transitionfunctor v ∗ : ( C /I ) ⊳ j ′ → ( C /I ⊳ ) j admits a left-adjoint v ! , and for any square (2.39) , the map v ◦ c → c ◦ v adjoint to the isomorphism c ◦ v ∗ ∼ = v ∗ ◦ c is itself an isomorphism. Then γ ⊳ is a cofibration.Proof. The first condition insures that γ ⊳ is a precofibration, with transitionfunctor f ! ∼ = v ! ◦ c ! for any map f = v ◦ c in J , where c is cocartesian and v is vertical over I . The second condition then insures that the maps (2.2)are isomorphisms, so that the precofibration γ ⊳ is a cofibration. (cid:3) Lemma 2.20.
Assume that π : J ⊳ → I has a fully faithful right-adjoint η : I → J ⊳ , and let C ′ = η ∗ ( C /J ) ⊳ . Then the restriction functor (2.40) η ∗ : Sec v ( J ⊳ , ( C /J ) ⊳ ) → Sec( I, C ′ ) is an equivalence.Proof. Consider the category ar v ( J ), with the projections s, t : ar v ( J ) → J and the evaluation functor ev : [1] × ar v ( J ) → J . Then ev ∗ C → [1] isa cofibration with fibers ( ev ∗ C ) ∼ = s ∗ C , ( ev ∗ C ) ∼ = t ∗ C and a transitionfunctor q : s ∗ C → t ∗ C cocartesian over ar v ( J ). Since t is a cofibration byExample 2.12, ar v ( J ) is cofibered over I , and we can consider the reflection C ♭ = ( ev ∗ C / [1] × ar v ( J )) ⊳ . We have ([1] × ar v ( J )) ⊳ ∼ = [1] o × ar v ( J ⊳ ), withthe projections s, t : ar v ( J ⊳ ) → J ⊳ , and C ♭ is fibered over [1] o = [1] withfibers C ♭ ∼ = s ∗ ( C /J ) ⊳ , C ♭ ∼ = t ∗ ( C /J ) ⊳ and the transition functor q ⊳ . Since C ♭ → [1] is a fibration, we also have the functor p : C ♭ → C ♭ right-adjoint tothe full embedding C ♭ ⊂ C ♭ given by id on C ♭ and q ⊳ on C ♭ . We now observethat t : ar v ( J ⊳ ) → J ⊳ has a fully faithful right-adjoint ε : J ⊳ → ar v ( J ⊳ )sending j ∈ J ⊳ to the adjunction map j → η ( π ( j )), we have s ◦ ε ∼ = id and t ◦ ε ∼ = η ◦ π , so that ε ∗ C ♭ ∼ = ( C /J ) ⊳ , ε ∗ C ♭ ∼ = π ∗ C ′ , and p restricts to a functor ε ∗ ( p ) : ε ∗ C ♭ → ( C /J ) ⊳ . It then induces a functor ε ∗ ( p ) ∗ : Sec( J ⊳ , ( C /J ) ⊳ ) → Sec([1] × J ⊳ , ε ∗ C ♭ )sending a section σ to a triple h σ , σ , α i of sections σ ∈ Sec( J ⊳ , ( C/J ) ⊳ ), σ ∈ Sec( J ⊳ , π ∗ C ′ ) and a map α : σ → ε ∗ ( q ⊳ ) ◦ σ . We have σ ∼ = σ and σ ∼ = π ∗ η ∗ σ . Moreover, ε ∗ ( q ⊳ ) ◦ σ is cartesian along v , and α is anisomorphism if and only if so is σ = σ . Therefore ε ∗ ( q ⊳ ) ◦ π ∗ is right-adjointto η ∗ , and provides an inverse equivalence to (2.40). (cid:3) I equipped with a factorization system h L, R i , consider the cofibration t : ar R ( I ) → I of Example 2.12, and let tw R ( I ) = ar R ( I ) ⊳ . The cocartesian functor ar ( I ) → ar R ( I ) left-adjoint tothe embedding ar ( R )( I ) ⊂ ar ( I ) induces a functor(2.41) π R : tw ( I ) ∼ = ar ( I ) ⊳ → tw R ( I ) , and for any two cofibrations C , C ′ → I as in Example 2.18, we can considerthe cofibration(2.42) F un L ( C /I, C ′ /I ) = π R ∗ ( s o × t ) ∗ Fun I ( C , C ′ ) → tw R ( I ) . Then (2.25) and (2.36) provide an equivalence(2.43) Fun LI ( C , C ′ ) ∼ = Sec t ∗ L ( tw R ( I ) , F un L ( C /I, C ′ /I )) . Explicitly, we have a factorization system h t ∗ L ∩ c, t ∗ R i on ar R ( I ), and (2.41)is a fibration over t ∗ R , so that for any object in ar R ( I ) represented by anarrow r : i ′ → i , the embedding(2.44) tw ( i \ L I ) ∼ = r \ t ∗ L ∩ c tw ( I ) ⊂ r \ tw ( I )is right-admissible by Example 1.6. This gives a framing for the functoropposite to (2.41), and if we use this framing to compute π L ∗ and apply(2.36) for the comma-fiber i \ L I , we obtain an identification(2.45) F un L ( C /I, C ′ /I ) r ∼ = Fun ∀ i \ L I (( f ∗ ) ∗ p ∗ i ′ C , p ∗ i C ′ ) , where f ∗ : i \ L I → i ′ \ L I stands for the transition functor of the fibration s : ar L ( I ) → I . In particular, if L = Iso , then F un L ( C /I, C ′ /I ) is the product Fun I ( C , C ′ ) × I o × I tw ( I )), while at the other extreme, if L = ∀ , then (2.42)coincides with (2.29).We can then consider the reflection ( F un L ( C /, C ′ /I ) / tw R ( I )) ⊳ of thecofibration (2.42). By definition, it is cofibered over I and comes equippedwith a functor(2.46) γ : ( F un L ( C /I, C ′ /I ) / tw R ( I )) ⊳ → ar R ( I ) ∼ = ( tw R ( I )) ⊳ cocartesian over I . Moreover, the projection t : ar R ( I ) → I has a fullyfaithful right-adjoint η : I → ar R ( I ) induced by (1.7) that is cocartesianover I L . Thus if we let(2.47) Fun L ( C /I, C ′ /I ) = η ∗ ( F un L ( C /I, C ′ /I ) / tw R ( I )) ⊳ → I, L , and (2.38) and Lemma 2.20 providean equivalence(2.48) Sec t ∗ L ( tw R ( I ) , F un L ( C /I, C ′ /I )) ∼ = Sec L ( I, Fun L ( C /I, C ′ , I )) . Combining (2.48) with (2.43) then gives an equivalence(2.49) Fun LI ( C , C ′ ) ∼ = Sec L ( I, Fun L ( C /I, C ′ /I )) . If L = ∀ , this is the equivalence (2.30) of Example 2.15.The “favourable circumstances” that we have mentioned are those whenthe functor (2.46) satisfies the assumptions of Lemma 2.19. Then it is acofibration over the whole I , and so is its restriction (2.47). A pre-order on a set is a binary relation ≤ that is transitiveand reflexive but not necessarily antisymmetric. Equivalently, a pre-orderedset is a small category that has at most one morphism between any twoobjects. We note that for any pre-ordered set J , the categories J o , J < , J > are also pre-ordered sets. The two opposite examples of a pre-order are themaximal one (a morphism between any two objects) or a partial order inthe usual sense. For every n ≥
0, we denote by [ n ] the partially ordered set { , . . . , n } , with the standard order (so that we have [ n ] < ∼ = [ n ] > ∼ = [ n +1]).In particular, [0] is the point category pt , and [1] is the single arrow categoryof Subsection 1.1. We denote by V = { , } < the partially ordered set withthree elements 0 , , o , and order relations 0 , ≥ o . Any set S with themaximal pre-order is denoted e ( S ). More generally, for any preordered set J , define a J -augmented set as a set S equipped with a map π : S → J ; thenfor any such h S, π i , we let e ( S/J ) be the set S with the pre-order inducedby π – that is, s ≤ s ′ iff π ( s ) ≤ π ( s ′ ) (in particular, e ( S/ pt ) ∼ = e ( S ), and e ( J/J ) ∼ = J ). We say that a J -augmented set S is proper if π : S → J issurjective and we note that for a proper J -augmented set S , e ( S/J ) → J isan equivalence of categories.We denote by Γ the category of finite sets, and we let Γ + be the categoryof finite sets and partially defined maps between them – that is, a map from S to S in Γ + is a diagram(3.1) S i ←−−−− e S f −−−−→ S in Γ with injective i . Equivalently, Γ + is the category of finite pointedsets, with the equivalence sending S to its union S + = S ∪ { o } with an35dded distinguished element. For clarity, we will always denote a finite set S considered as an object in Γ + by S + . A map in Γ + is an anchor map resp. a structural map if f resp. i in the diagram (3.1) is an isomorphism.The category Γ + is pointed, with the initial and terminal object o = ∅ + consisting of a single distinguished element, and coproducts in Γ are alsocoproducts in Γ + , traditionally written as ( S ⊔ S ′ ) + = S + ∨ S ′ + . For any S, S ′ ∈ Γ, the embeddings i : S → S ⊔ S ′ , i ′ : S ′ → S ⊔ S ′ define anchormaps(3.2) a : S + ∨ S ′ + → S + , a ′ : S + ∨ S ′ + → S ′ + given by the diagrams (3.1) with f = id and i = i resp. i = i . Cartesianproduct of finite sets is functorial with respect to the diagrams (3.1), thusdefines a product functor(3.3) m : Γ + × Γ + → Γ + . In terms of pointed sets, m ( S + × S ′ + ) is the usual smash product S + ∧ S ′ + =( S + × S ′ + ) / (( S + × o ) ∪ ( o × S ′ + )). As usual, we denote by ∆ the category formed by non-empty finite ordinals [ n ], n ≥
0, and order-preserving maps between them.Note that for any [ n ], we have a unique isomorphism [ n ] o ∼ = [ n ], and sending[ n ] to [ n ] o gives an involution ι : ∆ → ∆. Adding the empty ordinal ∅ to∆ gives the category ∆ < , with o = ∅ (for consistency, we will also denoteit [ − V : ∆ < → Sets the forgetful functor. For any map g : [ n ] → [ m ] in ∆ < with non-empty [ m ], we can form the cartesian squareof partially ordered set(3.4) [ n ] g e g −−−−→ [ n ] y y g V ([ m ]) −−−−→ [ m ] , where V ([ m ]) is equipped with the discrete order, and then [ n ] g is the disjointunion of ordinals [ n v ] indexed by v ∈ V ([ m ]) = { , . . . , m } (non-empty if g issurjective, possibly empty otherwise). This provides a canonical equivalence(3.5) ∆ < / [ m ] ∼ = ∆ < ( m +1) between the fiber category ∆ < / [ m ] and the product of m + 1 copies of ∆ < .36or any n ≥ l ≥
0, we denote by s, t : [ l ] → [ n ] the (unique) embeddingsonto the initial resp. terminal segment of the target ordinal, and we notethat we have a cocartesian square(3.6) [0] t −−−−→ [ l ] s y y s [ n − l ] t −−−−→ [ n ] . The maps s and t for different n ≥ l ≥ s , ∆ t ⊂ ∆ (as abstract categories, both are equivalent to the partially ordered set N of non-negative integers). We say that a map f : [ n ] → [ m ] is special resp. antispecial if f (0) = 0 resp. f ( n ) = m , and we denote by ∆ + , ∆ − ⊂ ∆ thedense subcategories defined by the classes of special resp. antispecial maps.We note that we have ∆ s ⊂ ∆ + and ∆ t ⊂ ∆ − . A map is bispecial if it isboth special and antispecial, and ∆ ± = ∆ + ∩ ∆ − ⊂ ∆ is the correspondingdense subcategory. A map f is special iff ι ( f ) is antispecial, so that theinvolution ι establishes a equivalences ι : ∆ + ∼ = ∆ − , ι : ∆ ± ∼ = ∆ ± .For another interpretation of special maps, consider the embedding ∆ → Cat sending an ordinal [ n ] to the opposite ordinal [ n ] o considered as a smallcategory, and let(3.7) ν q : ∆ q → ∆be the corresponding cofibration. Explicitly, ∆ q is the category of pairs h [ n ] , l i , [ n ] ∈ ∆, l ∈ [ n ], with morphisms h [ n ] , l i → h [ n ′ ] , l ′ i given by maps f : [ n ] → [ n ′ ] such that f ( l ) ≤ l ′ . The projection ν q sends h [ n ] , l i to [ n ], ithas a right-adjoint ν † : ∆ → ∆ q sending [ n ] to h [ n ] , i , and this in turn hasa right-adjoint ν ⊥ : ∆ q → ∆ sending h [ n ] , l i to [ n − l ] = { l, . . . , n } ⊂ [ n ]. Thedense subcategory in ∆ q spanned by maps cocartesian over ∆ is equivalent tothe right comma-fiber [0] \ ∆, and a map f in ∆ is special if and only if ν † ( f )is cocartesian over ∆. For antispecial maps, there is a similar descriptionusing the cofibration ι ∗ ∆ q .The embedding functor ρ : ∆ + → ∆ admits a left-adjoint λ : ∆ → ∆ + sending [ n ] to [ n ] < . The composition κ = ρ ◦ λ is then an endofunctor of thecategory ∆, and we have the adjunction map(3.8) a : id → κ equal to the embedding t : [ n ] → [ n +1] = κ ([ n ]) on any [ n ] ∈ ∆. The functor λ , hence also κ , extends to ∆ < ⊃ ∆ by setting κ ( o ) = λ ( o ) = ∅ < = [0].37he embedding functor ρ ι = ι ◦ ρ ◦ ι : ∆ − → ∆ also admits a left-adjoint λ ι = ι ◦ λ ◦ ι : ∆ → ∆ − , with the composition κ ι = λ ι ◦ ρ ι ∼ = ι ◦ κ ◦ ι and the adjunction map a ι : id → κ ι , and the embedding ρ ♭ : ∆ ± → ∆admits a left-adjoint λ ♭ : ∆ → ∆ ± with the composition κ ♭ = λ ♭ ◦ ρ ♭ andthe adjunction map id → κ ♭ . We have κ ♭ ∼ = κ ◦ κ ι ∼ = κ ι ◦ κ , and we have afunctorial cocartesian square(3.9) id a −−−−→ κ a ι y y κ ι −−−−→ κ ♭ . Both κ ι and κ ♭ extend to ∆ < , with κ ι ( o ) = ∅ > = [0] and κ ♭ ( o ) = ∅ <> = [1].The concatenation [ m ] ◦ [ n ] ∼ = [ m + n ] of two ordinals [ m ] , [ n ] ∈ ∆ is theirdisjoint union [ m ] ⊔ [ n ] ordered left-to-right. Concatenation is functorial withrespect to either variable, and we have κ ([ n ]) ∼ = [0] ◦ [ n ], κ ι ([ n ]) ∼ = [ n ] ◦ [0]and κ ♭ ([ n ]) ∼ = [0] ◦ [ n ] ◦ [0].A map f : [ n ] → [ m ] in ∆ considered as a functor between small cate-gories has a right-adjoint f † : [ m ] → [ n ] if and only if it is special, and inthis case f † is antispecial; for left-adjoints f † , the situation is dual. Thussending f to its right resp. left-adjoint provides equivalences(3.10) ∆ + ∼ = ∆ o − , ∆ − ∼ = ∆ o + . Moreover, for any map f in ∆ < , not only λ ( f ) but also the right-adjoint λ ( f ) † is special. Thus λ ( f ) † is actually bispecial, and composing λ with theequivalence (3.10) gives a functor(3.11) ∆ < → ∆ o ± that also happens to be an equivalence. Restricting to ∆ ⊂ ∆ < , we obtaina canonical functor(3.12) θ : ∆ → ∆ o . We say that a map f : [ n ] → [ m ] in ∆ is a left-anchor map resp. right-anchor map if f = s resp. f = t , and we say that f is an anchor map iff f : [ n ] → [ m ] identifies [ n ] with some segment { l, l +1 , . . . , l + n } ⊂ [ m ] of theordinal [ m ], or equivalently, if f = s ◦ t for some right-anchor t : [ n ] → [ n + l ]and left-anchor s : [ n + l ] → [ m ].By abuse of terminology, we say that a map f in ∆ o is special, anti-special, bispecial, anchor, left-anchor or right-anchor if so is the oppositemap f o in ∆. 38he classes of bispecial and anchor maps form a factorization system h± , a i in ∆ in the sense of Definition 1.1, and we also have factorizationsystems h + , t i , h− , s i . In particular, we have the fibrations and cofibrationsof Example 2.12. Since ∆ s ∼ = ∆ t ∼ = N , we have ar s (∆) ∼ = ar t (∆) ∼ = ∆ q ,and the cofibrations ar s (∆) , ar t (∆) → ∆ are both identified with (3.7). Thefibrations ar (∆) , ar + (∆) , ar − (∆) , ar ± → ∆ have a useful universl propertywith respect to cocartesian square (3.6). For any such square, the targetof any map f : [ n ] → [ n ′ ] fits into a similar square with l ′ = f ( l ); this isfunctorial in f and defines a cartesian diagram(3.13) ar (∆) [ n ] −−−−→ ar + (∆) [ n − l ] y y ar − (∆) [ l ] −−−−→ ar ± (∆) [0] , where of course ar ± (∆) [0] ∼ = pt , so that we actually have a decomposition ar (∆) [ n ] ∼ = ar + (∆) [ n − l ] × ar − (∆) [ l ] . If f ∈ ar (∆) [ n ] is special resp. antispe-cial, then so are both of the components of its decomposition, so that inparticular, (3.13) induces a decompositon(3.14) ar ± (∆) [ n ] ∼ = ar ± (∆) [ l ] × ar ± (∆) [ n − l ] , where projections onto both factors are given by the transition functors ofthe fibration s : ar ± (∆) → ∆.Yet another useful factorization system h p, i i in ∆ is formed by the classes p resp. i of surjective resp. injective maps. We have an embedding ar p (∆) ⊂ ar ± (∆) cartesian with respect to the fibration s : ar p (∆) → ar p (∆), and(3.14) induces a corresponding decomposition for the fibration ar p (∆). Anysurjective map p : [ n ] → [ m ] considered as a functor between small categoriesis both a fibration and a cofibration, and for any injective map i : [ l ] → [ m ],there exists a cartesian square(3.15) [ q ] i ′ −−−−→ [ n ] p ′ y y p [ l ] i −−−−→ [ m ]in ∆ with some [ q ], injective i ′ and surjective p ′ (this is obvious from (3.5)). We denote by ∆ o Sets the category of simplicial sets (thatis, functors from ∆ o to Sets). For any simplicial set X , its category of implices ∆ X is the corresponding discrete fibration ∆ X → ∆. Explicitly,its objects are pairs h [ n ] , x i , [ n ] ∈ ∆, x ∈ X ([ n ]), and morphisms from h [ n ] , x i to h [ n ′ ] , x ′ i are given by maps f : [ n ] → [ n ′ ] such that X ( f o )( x ′ ) = x .We let ∆ o X = (∆ X ) o = ∆ o [ X ] be the opposite category. The augmentedcategory of simplices is given by ∆ < X = (∆ X ) < , with the opposite category∆ o> X = (∆ < X ) o = (∆ X ) o> .Analogously, we denote by Sets + the category of pointed sets, and we let∆ o Sets + be the category of simplicial pointed sets. One distinguished sim-plicial pointed set is the standard simplicial circle Σ obtained by taking thestandard simplicial interval ∆ , ∆ ([ n ]) = ∆([ n ] , [1]), and gluing togetherthe two ends. The simplicial set Σ is finite, thus factors through a functor(3.16) Σ : ∆ o → Γ + . A map f in ∆ is an anchor map iff Σ( f o ) is an anchor map in Γ + .A contraction of an augmented simplicial object c > : ∆ o> → C insome category C is a functor c + : ∆ o + → C equipped with an isomorphism λ o ∗ c + ∼ = c > . Any simplicial set X : ∆ o → Sets has a tautological augmenta-tion X > sending o ∈ ∆ o> to the one-point set pt , and we say that X is con-tractible if X > admits a contraction X + : ∆ o + → Sets. By the Grothendieckconstruction, such a contraction defines a discrete fibration ∆ + X + → ∆ + with the opposite discrete cofibration ∆ o + X + = (∆ + X + ) o → ∆ o + , and λ extends to an embedding ∆ < X → ∆ + X + . Definition 3.1.
For any contractible simplicial set X ∈ ∆ o Sets, an aug-mentation c > : ∆ o> X → C of a functor c : ∆ o X → C to some category C is contractible if it further extends to ∆ o + X + ⊃ ∆ o> X for some contraction X + of the set X . Lemma 3.2.
For any contractible simplicial set X and category C , a locallyconstant functor c : ∆ o X → C is constant, and a contractible augmentation c > of a functor c : ∆ o X → C is exact.Proof. The embedding λ X : ∆ X → ∆ < X → ∆ + X + admits a left-adjoint ρ X : ∆ + X + → ∆ X given by the composition∆ + X + a ∗ −−−−→ ρ ∗ λ ∗ ∆ + X + ∼ = ρ ∗ ∆ X −−−−→ ∆ X, where a : λ ◦ ρ → id is the adjunction map. Therefore ∆ o X ⊂ ∆ o + X + is aleft-admissible subcategory. On the other hand, by definition, the set X + ( o )consists of a single point, say x , and then h o, x i ∈ ∆ + X + is the terminal40bject, thus defines a left-admissible embedding pt → ∆ o + X + . Then (1.14)immediately shows that a contractible augmentation is exact. Moreover,any locally constant c + : ∆ o + X + → C is constant, and any locally constant c : ∆ o X → C inverts the adjunction map id → ρ oX ◦ λ oX , so that c ∼ = λ o ∗ X ρ o ∗ X c ,and c is constant as well. (cid:3) As usual, the nerve N ( I ) of a small category I is the simplicial set suchthat N ( I )([ n ]), [ n ] ∈ ∆ is the set of functors i q : [ n ] → I . If I is connected, itsnerve N ( I ) is connected, and if I has an initial object, N ( I ) is contractible.Indeed, since [0] ∈ ∆ + is the initial object, we have a natural projection ρ o ∗ N ( I ) → N ( I )([0]), or in other words, a decomposition(3.17) ρ o ∗ N ( I ) = a i ∈ I N ( I ) i , N ( I ) i : ∆ o + → Sets . Moreover, for any i ∈ I , we have λ o ∗ N ( I ) i ∼ = N ( i \ I ). But if i ∈ I is aninitial object, the map N ( i \ I ) → N ( I ) induced by the forgetful functor i \ I → I is an isomorphism, so that N ( I ) i is a contraction of N ( I ).The nerve is functorial in I and provides a fully faithful embeddingCat → ∆ o Sets from the category Cat of small categories to the category ofsimplicial sets. Its essential image consists of simplicial sets X : ∆ o → Setsthat send cocartesian squares (3.6) to cartesian squares in Sets (this is calledthe
Segal condition ). The nerve embedding commutes with limits but notwith colimits. Colimits in Cat exist but are hard to compute except in somespecial cases. Here is one obvious example.
Example 3.3.
If we have full subcategories I , I ⊂ I that are left-closedin the sense of Example 1.15, then I ∪ I is left-closed in I , I = I ∩ I isleft-closed in I , I and I , and we have I ∪ I ∼ = I ⊔ I I . Giving a functor I ∪ I → E to some category E is equivalent to giving functors γ : I → E , γ : I → E , and an isomorphism between their restrictions to I .Another example is path categories of quivers. Namely, a quiver Q is acollection of two sets V ( Q ), E ( Q ) of its vertices and edges , and two maps s, t : V ( Q ) → E ( Q ) sending an edge to its source resp. target. Equivalently,a quiver is a functor Q : D o → Sets, where D is the category with twoobjects 0, 1 and two non-trivial maps s, t : 0 →
1, and we take V ( Q ) = Q (0), E ( Q ) = Q (1). We have the embedding δ : D → ∆ sending 1 to [1], 0 to [0]and s , t to the maps s , t in ∆, and the opposite functor δ o factors as(3.18) D o α −−−−→ ∆ oa β −−−−→ ∆ o , a ⊂ ∆ is the dense subcategory spanned by anchor maps. Let D o Sets be the category of quivers, and let ∆ oa Sets be the category of functorsfrom ∆ oa sets. Then the composition P = β ! α ∗ : D o Sets → ∆ o Sets ofthe right Kan extension α ∗ and the left Kan extension β ! factors throughCat ⊂ ∆ o Sets and is left-adjoint to the restriction functor δ o ∗ : Cat ⊂ ∆ o Sets → D o Sets. For any Q ∈ D o Sets, P ( Q ) is called the path category ofthe quiver Q . By adjunction, the functor P : D o Sets → Cat commutes withcolimits, and colimits of quivers can be computed pointwise.
Remark 3.4.
The presense of two different Kan extensions in the con-struction of the path category P ( Q ) is easy to understand if one observesthat for any n ≥
0, [ n ] is the path category of a “string quiver” [ n ] δ with V ([ n ] δ ) = V ([ n ]) and edges e ∈ E ([ n ] δ ) = { , . . . , n } corresponding to inter-vals { l − , l } ⊂ [ n ]. Then maps between such quivers are exactly the anchormaps, the squares (3.6) are induced by cocartesian squares of quivers, sothat the Segal condition makes sense for objects X ∈ ∆ oa Sets, and α ∗ in(3.18) identifies D o Sets with the category of functors X ∈ ∆ oa Sets satifyingthis condition. Explicitly, it is given by α ∗ Q ([ n ]) ∼ = Hom D o Sets ([ n ] δ , Q ) , [ n ] ∈ ∆ . The Kan extension β ! can be then computed by (2.12) and the decomposi-tion (2.16) for the anchor/bispecial factorization system on ∆ o , where oneobserves that since all isomorphisms in ∆ o are identity maps, the cofibration ar ± (∆ o ) ♮ → ∆ o is discrete. It is easy to see from (3.14) that β ! preservesthe Segal condition, and this creates the adjunction between P and δ o ∗ . Among other things, Lemma 3.2 allows to prove someuseful results above nerves of small categories. To simplify notation, for anysmall I , denote ∆ I = ∆ N ( I ), and similarly for ∆ o , ∆ < and ∆ o> . Note that∆ I o ∼ = ι ∗ (∆ I ), where ι : ∆ → ∆ sends [ n ] to [ n ] o . Example 3.5.
For any [ n ] ∈ ∆ considered as a small category, we have∆[ n ] ∼ = ∆ / [ n ], ∆ < [ n ] ∼ = ∆ < / [ n ] and ∆ o [ n ] ∼ = [ n ] \ ∆ o , ∆ o> [ n ] ∼ = [ n ] \ ∆ o> .As we saw, if I has an initial object i ∈ I , then the nerve N ( I ) iscontractible, so that by Lemma 3.2, ∆ o I is simply connected in the sense ofDefinition 1.7. The first corollary of this is the following standard fact. Lemma 3.6.
The diagonal embedding ∆ o → ∆ o × ∆ o is cofinal in the senseof Definition 1.16. roof. For each [ n ] × [ m ] ∈ ∆ o , the right comma-fiber ([ n ] × [ m ]) \ ∆ o is∆ o ([ n ] × [ m ]) by Example 3.5, and since [ n ] × [ m ] has an initial object, it issimply connected by Lemma 3.2. (cid:3) Remark 3.7.
A standard implication of Lemma 3.6 is that if a cocompletecategory E has finite products, then colim ∆ o commutes with these products. Lemma 3.8.
Assume given a full embedding γ : I ′ → I of small categoriesthat is either left or right-closed in the sense of Example 1.15. Then theinduced functor ∆ o> ( γ ) : ∆ o> I ′ → ∆ o> I admits a left-adjoint functor (3.19) µ : ∆ o> I → ∆ o> I ′ , and this functor is a cofibration.Proof. By Example 3.5 and (3.5), we have ∆ o> [1] ∼ = ∆ o> × ∆ o> , and thenby Example 1.2, the classes π ∗ Iso and π ∗ Iso form a factorization system on∆ o> [1], in either order. Let us denote them by 0 and 1 and call them 0 -special and 1 -special maps (so that l -special maps are those that are isomorphismsover l ∈ [1]). If γ is left-closed, then I ′ = I for a functor χ : I → [1], and∆ o> ( χ ) : ∆ o> I → ∆ o> [1] is a discrete cofibration. Thus if we say that a map f in ∆ o> I is l -special, l = 0 , o> ( χ )( f ), then 0-special and1-special maps also form a factorization system h , i on ∆ o> I . Considerthe cofibration t : ar (∆ o> I ) → ∆ o> of Example 2.12, and observe that∆ o> ( γ ) ∗ ar (∆ o> I ) ∼ = ∆ o> I . Then µ = ∆ o> ( γ ) ∗ t is our cofibration (3.19),and the adjunction is obvious. For a right-closed γ , interchange 0 and 1. (cid:3) Now take some [ n ] ∈ ∆, and assume given a small category I and afunctor π : I → [ n ], with the induced functor ∆ I → ∆[ n ] ∼ = ∆ / [ n ]. Considerthe full subcategories ∆ p / [ n ] ⊂ ∆ + / [ n ] ⊂ ∆ / [ n ] spanned by surjective resp.special arrows f : [ m ] → [ n ], and let(3.20) ∆ p ( I/ [ n ]) = ∆ I × ∆[ n ] ∆ p / [ n ] , ∆ + ( I/ [ n ]) = ∆ I × ∆[ n ] ∆ + / [ n ] . Say that π : I → [ n ] is an iterated cylinder if it is a cofibration with non-empty fibers. Lemma 3.9.
For any iterated cylinder π : I → [ n ] , ∆ p ( I/ [ n ]) ⊂ ∆ + ( I/ [ n ]) is cofinal. If I has an initial object i , then ∆ + ( I/ [ n ]) ⊂ ∆ I is also cofinal. roof. The second claim follows from Lemma 3.2 and Lemma 1.18: since I ∼ = i \ I , we have a left-admissible subcategory ∆ + N ( I ) i ⊂ ∆ I , and sincethe fiber I ⊂ I is non-empty, we have i ∈ I , so that ∆ + N ( I ) i ⊂ ∆ + ( I/ [ n ]).For the first claim, note that by definition, objects in ∆ I are pairs h [ n ] , i q i of an object [ n ] ∈ ∆ and a functor i q : [ n ] → I , l i l . For any l ∈ [ n ], let∆ l I ⊂ ∆ I be the full subcategory consisting of pairs h [ m ] , i q i such that theimage of the map π ◦ i q : [ m ] → [ n ] contains s ([ l ]) = { , . . . , l } ⊂ [ n ]. Then∆ p ( I/ [ n ]) = ∆ I , ∆ + ( I/ [ n ]) = ∆ n I , and by induction and Lemma 1.18, itsuffices to prove that ∆ l +1 I ⊂ ∆ l I is cofinal for any l = 0 , . . . , n −
1. Choosesuch an l , consider the embeddings s : [ l ] → [ n ], t : [ n − l − → [ n ], and denote I ≤ l = s ∗ I , I >l = t ∗ I . Then the full embedding I ≤ l → I is left-closed, andthe functor opposite to the cofibration (3.19) sends ∆ l I into ∆ I ≤ l ⊂ ∆ < I ≤ l .Therefore we have the induced fibration ∆ l I → ∆ I ≤ l , ∆ l +1 I ⊂ ∆ l I is asubfibration, and for any object h [ m ] , i q i ∈ ∆ I ≤ l , their fibers are(3.21) (∆ l +1 I ) h [ m ] ,i q i ∼ = ∆ + ( i m \ I >l ) ⊂ (∆ l I ) h [ m ] ,i q i ∼ = ∆( i m \ I >l ) . But since I → [ n ] is a cofibration, the iterated cylinder ( i m \ I >l ) / [ n − l − (cid:3) As an application of Lemma 3.9, consider the arrow categories ar p (∆), ar ± (∆), and treat the embedding δ : ar p (∆) o → ar ± (∆) o as a functor over∆ o with respect to the projections induced by t of (1.5). Then for any object c ∈ ar ± (∆) o , we have an embedding(3.22) ar p (∆) o / δt o ∗ Iso c ⊂ ar p (∆) o / δ c. Were δ cocartesian over ∆ o , the embedding (3.22) would have been left-admissible. While δ is certainly not cocartesian — its source and target arenot even cofibrations — we still have the following result. Lemma 3.10.
The embedding (3.22) is cofinal for any c ∈ ar ± (∆) o .Proof. By Example 1.3, the bispecial/anchor factorization system h± , a i on∆ induces a factorization system on ar ± (∆), so that any map c → c ′ in ar ± (∆) uniquely factors as c l −−−−→ c ′′ r −−−−→ c ′ with l ∈ ( s × t ) ∗ ± and r ∈ ( s × t ) ∗ a . One immediately checks that if c ′ lies in ar p (∆) ⊂ ar ± (∆), then so does c ′′ . Therefore by Example 1.5 and44xample 1.6, if we denote by δ ′ : ar p (∆ ± ) o → ar (∆ ± ) the restriction of theembedding δ to ar p (∆ ± ) o ⊂ ar p (∆) o , then the embeddings ar p (∆ ± ) o / δ ′ c ⊂ ar p (∆) o / δ c, ar p (∆ ± ) o / δ ′ t o ∗ Iso c ⊂ ar p (∆) o / δt ∗ Iso c are left-admissible. Then by Lemma 1.18, it suffices to prove the claim with(3.22) replaced by the embedding(3.23) ar p (∆ ± ) o / δ ′ t o ∗ Iso c ⊂ ar p (∆ ± ) o / δ ′ c. To do this, take some object c ′ ∈ ar p (∆ ± ) o / δ ′ c , and consider the rightcomma-fiber I ( c, c ′ ) = c ′ \ ( ar p (∆ ± ) o / δ ′ t ∗ Iso c )of the embedding (3.23). Explicitly, c is a bispecial map f : [ n ] → [ m ], c ′ isa pair of a surjective map f ′ : [ n ′ ] → [ m ′ ] and a map g : f → f ′ in ar ± (∆)with bispecial components g n : [ n ] → [ n ′ ], g m : [ m ] → [ m ′ ], and I ( c, c ′ ) isthen opposite to the category of commutative diagrams(3.24) [ n ] −−−−→ [ n ′′ ] −−−−→ [ n ′ ] f y y p y f ′ [ m ] [ m ] g m −−−−→ [ m ′ ]in ∆ ± with surjective p whose outer rectangle represents the map g . Wehave to prove that I ( c, c ′ ) is simply connected. Since bispecial maps admitthe decomposition (3.14) with respect to cartesian squares (3.6), the wholediagram (3.24) also functorially decomposes into diagrams for [ l ] and [ n − l ],so that by induction, it suffices to consider the case [ n ] = 1. Then the leftvertical arrow in (3.24) carries no information and can be forgotten, and weend up with an equivalence I ( c, c ′ ) ∼ = ∆ op ( g ∗ m [ n ′ ] / [ m ]) . But [ n ′ ] has an initial object, and since f ′ : [ n ′ ] → [ m ′ ] is surjective, it is acofibration and an iterated cylinder. Then g ∗ m f ′ : g ∗ m [ n ′ ] → [ m ] is also aniterated cylinder with an initial object, and we are done by Lemma 3.9 andLemma 3.2. (cid:3) For any integer n ≥
1, we denote by [ n ] Λ the category whoseobjects are residues a ∈ Z /n Z , with maps from a to a ′ given by integers l ≥ a ′ = a + l mod n (equivalently, [ n ] Λ is the path category45f the wheel quiver [ n ] λ with n vertices, and l is the length of the path).For any a ∈ [ n ] Λ , we have a map τ a : a → a given by n , and for anyfunctor f : [ n ] Λ → [ m ] Λ , there exists a unique integer deg f ≥ f ( τ a ) = τ deg ff ( a ) for all a ∈ [ n ] Λ . We say that f is non-degenerate if deg f ≥ horizontal if deg f = 1. The cyclic category Λ of A. Connes is thecategory with objects [ n ], n ≥
1, and morphisms from [ n ] to [ m ] given byhorizontal functors [ n ] Λ → [ m ] Λ . For any [ n ] ∈ Λ, we denote by V ([ n ] λ )the set of objects or the category [ n ] Λ (or equivalently, the set of vertices ofthe corresponding wheel quiver). Every horizontal functor f : [ n ] Λ → [ m ] Λ admits a left-adjoint f † : [ m ] Λ → [ n ] Λ , and sending [ n ] to [ n ] and f to f o † provides an equivalence of categories(3.25) θ : Λ ∼ = Λ o . For any [ n ] in Λ and a horizontal functor f : [ n ] Λ → [1] Λ , we have a cartesiansquare of small categories(3.26) [ n − ε ( f ) −−−−→ [ n ] Λ y y f [0] ε −−−−→ [1] Λ , where [0] , [ n − ∈ ∆ are considered as small categories in the usual way,and ε in the bottom line is the embedding onto the unique object in [1] Λ (in terms of quivers, (3.26) corresponds to removing one edge from a wheelquiver to obtain a string quiver). Sending the rightmost colum in (3.26)to the leftmost column defines a functor Λ / [1] → ∆ that happens to be anequivalence; composing the inverse equivalence with the forgetful functorΛ / [1] → Λ gives an embedding j : ∆ → Λ. More generally, for any [ n ] ∈ Λ,we can choose a map f : [ n ] → [1], and this gives an identification(3.27) Λ / [ n ] ∼ = ∆ / [ n − . Passing to the opposite categories and identifying Λ ∼ = Λ o by (3.25), weobtain a dual embedding j o : ∆ o → Λ and an equivalence Λ \ [ n ] ∼ = ∆ o \ [ n ]dual to (3.27) for any [ n ] ∈ Λ. One can also see the equivalence [1] \ Λ ∼ = ∆ o directly in terms of the equivalence (3.11): if we let ω : [1] → [1] Λ be thefunctor that sends 0 , ∈ [1] to the unique object 0 ∈ [1] Λ , and the map 0 → τ : 0 →
0, then for any injective bispecial map b : [1] → [ n ] in46, we have the cocartesian square of small categories(3.28) [1] ω −−−−→ [1] Λ b y y b ′ [ n ] ω n −−−−→ [ n ] Λ , and the functor b ′ is horizontal (to see that (3.28) is cocartesian, note thatit is induced by a cocartesian square that glues the two end vertices of astring quiver to obtain a wheel). The correspodence b b ′ is functorial andidentifies [1] \ Λ with the full subcategory ∆ o ⊂ ∆ ± spanned by injectivebispecial maps. The identifications given by (3.26) and (3.28) intertwine theduality (3.25) and the embedding (3.12). Lemma 3.11.
For any cocomplete target category E , the functor Tw Λ of (2.13) takes values in Fun ∀ (Λ , E ) ∼ = Fun ∀ (Λ o , E ) ⊂ Fun(Λ o , E ) .Proof. For any E ∈ Fun(Λ , E ), E ′ = Tw Λ ( E ) exists since E is cocomplete,and we have to check that for any map f : [ m ] → [ n ] in Λ, E ′ inverts f o . Itsuffices to choose a map g : [1] → [ m ] and prove that E ′ inverts g o and ( f ◦ g ) o , so we may assume right away that [ m ] = 1. Then by (2.11), it suffices tocheck that the transition functor f o ! : tw (Λ o ) [ n ] → tw (Λ o ) [1] of the cofibration t : tw (Λ o ) → Λ o is cofinal. But by (3.27) and (3.25), this transition functorcan be identified with the projection (∆ / [ n − o ∼ = ∆ o [ n − → ∆ o , and thenits right comma-fibers are of the form ∆ o ([ n − × [ l ]), [ l ] ∈ ∆ o . As in theproof of Lemma 3.6, these are simply connected by Lemma 3.2. (cid:3) The cyclotomic category Λ R ⊂ Λ has the same objects [ n ], n ≥
1, and allnon-degenerate functors f : [ n ] Λ → [ m ] Λ as morphisms. A morphism f inΛ R is vertical if it is a discrete bifibration. Any morphism f in Λ R factorsas f = v ◦ h with horizontal h and vertical v , and h h, v i is a factorizationsystem on Λ R . For any integer l ≥
1, one defines the category Λ l as thecategory of vertical arrows v : [ nl ] → [ n ] in Λ R of degree l , with morphismsgiven by commutative square [ nl ] v −−−−→ [ n ] h ′ y y h [ n ′ l ] v ′ −−−−→ [ n ′ ]in Λ R with horizontal h , h ′ . Such a square is automatically cartesian. Send-ing an arrow to its source resp. target gives two functors(3.29) i l , π l : Λ l → Λ . i l is known as the edgewise subdivision functor and goes backto [S1]. The functor π l has no special name; it is a bifibration whose fibersare connected groupoids pt l with one object o , Aut( o ) = Z /l Z . We denote by Top + the category of pointedcompactly generated topological spaces, and we denote by Ho its homo-topy category obtained by localizing Top + with respect to the class of weakequivalences. For any bounded category I , we denote by Ho( I ) the local-ization of the category Fun( I, Top + ) with respect to the class of pointwiseweak equivalences. For any bounded functor γ : I → I ′ between boundedcategories, the pullback γ ∗ preserves weak equivalences, thus descends to afunctor γ ∗ : Ho( I ′ ) → Ho( I ). This functor γ ∗ has a left and a right-adjoint γ ! , γ ∗ : Ho( I ) → Ho( I ′ ) known as the homotopy Kan extensions . If I ′ is thepoint category pt , then γ ! = hocolim I is the homotopy colimit over I , and γ ∗ = holim I is the homotopy limit.For any bounded cofibration C → I , the transition functors ( f ! ) ∗ ofthe transpose fibration Fun( C /I, Top + ) ⊥ → I are pullback functors, sothat Fun( C /I, Top + ) ⊥ descends to a fibration H o ( C /I ) → I with fibers H o ( C /I ) i = Ho( C i ), i ∈ I . This fibration is also a cofibration, with transi-tion functors given by homotopy Kan extensions ( f ! ) ! . Any cocartesian func-tor γ : C ′ → C between two cofibrations over I induces a cartesian pullbackfunctor γ ∗ : H o ( C /I ) → H o ( C ′ /I ) that has a left-adjoint γ ! : Ho( C ′ /I ) → Ho( C /I ); for each i ∈ I , we have ( γ ∗ ) i ∼ = γ ∗ i and ( γ ! ) i ∼ = γ i ! . For any boundedcofibration C → I , we have a comparison functor(4.1) H o : Ho( C ) → Sec( I, Ho( C /I ))that reduces to Ho( C ) → Fun( I, Ho) if C = I . This functor is conservativeand commutes with pullbacks (but it is certainly not an equivalence unless I is a point).To compute homotopy Kan extensions, one actually needs to know verylittle about them (apart from their sheer existence known from e.g. [DHKS]that can be used as a black box). Namely, just as in the non-homotopicalcase, an adjunction between functors γ : I ′ → I , γ : I → I ′ induces anadjunction between the pullback functors γ ∗ and γ ∗ , and this yields (1.14)(with colimits replaced by homotopy colimits). Moreover, for a discrete cofi-bration π : I ′ → I , the left Kan extension π ! : Fun( I ′ , Top + ) → Fun( I, Top + )48espects weak equivalences, thus descends to the left homotopy Kan exten-sion, and dually for discrete fibrations and π ∗ . Together with (1.14), thisdescribes Kan extensions with respect to embeddings pt → I , and the de-scription immediately yields homotopical versions of (1.13), Lemma 1.14and the base change isomorphism (2.11) (where again, one replaces colimwith hocolim in all the statements, and uses γ ! to denote the homotopy Kanextension).For any bounded I and a class of maps v in I , we denote by Ho v ( I ) ⊂ Ho( I ) the full subcategory spanned by X ∈ Ho( I ) such that H o ( X ) lies inFun v ( I, Ho). We say that X ∈ Ho( I ) is locally constant if so is H o ( X ), sothat Ho ∀ ( I ) ⊂ Ho( I ) is the full subcategory spanned by locally constant X .For any functor γ : I ′ → I and any class v of maps in I , (1.10) induces afunctor(4.2) γ ∗ v : Ho v ( I ) → Ho γ ∗ v ( I ′ ) , and as in Definition 1.7, we say that γ is a homotopy localization if (4.2)is an equivalence for any v . We note that the localizations of Example 1.9are also homotopy localizations. Somewhat weaker, we say that γ is a weakequivalence if (4.2) is an equivalence for the class v = ∀ of all maps. Suchan equivalence is automatically compatible with limits and colimits: for any X ∈ Ho ∀ ( I ′ ), the natural maps(4.3) hocolim I ′ γ ∗ X → hocolim I X and(4.4) holim X → holim I ′ γ ∗ X are both isomorphisms in Ho. A homotopy localization is trivially a weakequivalence, and so are its one-sided inverses; in particular, it applies to leftand right-admissible full embeddings.In general, a functor γ : I ′ → I between small categories I , I ′ is a weakequivalence if and only if so is the induced map | I ′ | → | I | of the geometricrealizations of their nerves. However, this is a serious theorem that wewill not need. We restrict ourselves to the following trivial but very usefulobservation whose effectiveness was demonstrated to us by L. Hesselholt. Lemma 4.1.
Assume given functors γ : I ′ → I , γ : I → I ′ betweenbounded categories I ′ and I equipped with a map between Id and γ ◦ γ anda map between Id and γ ◦ γ (in either direction). Then γ ∗ ∀ and γ ∗ ∀ aremutually inverse equivalences. roof. A one-sided inverse to a weak equivalence is trivially a weak equiv-alence. For any I , consider the product I × [1], with the embeddings s, t : I → I × [1] onto I × I ×
1, and their common one-sided in-verse e : I × [1] → I given by the projection onto the first factor. Then e is aweak equivalence by Example 1.9, and therefoe so are s and t . Therefore forany two functors ϕ , ϕ : I ′ → I , a map ϕ → ϕ induced an isomorphism ϕ ∗ ∀ ∼ = ϕ ∗ ∀ . In the assumptions of the Lemma, this provides isomorphisms Id ∼ = γ ∗ ∀ ◦ γ ∗ ∀ and Id ∼ = γ ∗ ∀ ◦ γ ∗ ∀ . (cid:3) We will say that a bounded category I is homotopy contractible if thetautological projection τ : I → pt is a weak equivalence, and we will saythat a functor γ : I ′ → I between bounded categories is homotopy cofinal if its right comma-fiber i \ γ I ′ is homotopy contractible for any i ∈ I . Forany contractible I , holim I and hocolim I are isomorphic, and both are inverseequivalences to τ ∗∀ . As for (1.14), by virtue of (the dual homotopical versionof) (1.13), this implies that γ ∗ ◦ τ ∗ ∼ = ( τ ◦ γ ) ∗ for any cofinal functor γ : I ′ → I ,and then by adjunction, this in turn implies that (4.3) is an isomorphismfor any X ∈ Ho( I ).A contractible I is simply connected in the sense of Definition 2.10,and a homotopy cofinal full embedding γ is cofinal in the sense of Defini-tion 1.18. The converse is not true; however, Lemma 1.18, Remark 1.17 andall the criteria of cofinality of Section 1 still hold for homotopy cofinal em-beddings, and all the cofinal embeddings of Section 3 are homotopy cofinal.In particular, this is true for the embedding of Lemma 3.6, so that for any X ∈ Ho(∆ o × ∆ o ), the natural map(4.5) hocolim ∆ o δ ∗ X → hocolim ∆ o × ∆ o X induced by the diagonal embedding δ : ∆ o → ∆ o × ∆ o is a homotopyequivalence. This implies that hocolim ∆ o commutes with finite products. We begin by recalling the standard Segal con-struction of [S2] (with slightly different notation). As in Subsection 3.1, letΓ + be the category of finite pointed sets. A Γ -space is an object X ∈ Ho(Γ + ). Definition 4.2.
A Γ-space X is special if X ( o ) is contractible, and for any S, S ′ ∈ Γ, the map(4.6) X ( a ) × X ( a ′ ) : X ( S + ∨ S ′ + ) → X ( S + ) × X ( S ′ + )induced by the anchor maps (3.2) is a weak homotopy equivalence.50t is well-known that any special Γ-space X is a commutative monoidobject in Ho(Γ + ). Indeed, if we take S = S ′ and let d : Γ + → Γ + bethe functor sending S + to S + ∧ S + , then the anchor maps (3.2) and thecodiagonal maps m : S + ∧ S + → S + give rise to a diagram X × X X ( a ) × X ( a ′ ) ←−−−−−−−− d ∗ X X ( m ) −−−−→ X, and the map on the left is an equivalence, so it can be inverted in Ho(Γ + ).This defines a product map(4.7) X × X → X in Ho(Γ + ), and one checks that it is associative and commutative. In par-ticular, π ( X ( pt + )) is a commutative monoid, and X is called group-like ifthis monoid is a group. Definition 4.3.
For any bounded category I , an object X ∈ Ho(Γ + × I ) is stable if for any i ∈ I , its restriction to Γ + × I is a group-like special Γ-space. Proposition 4.4.
For any bounded category I , denote by Ho st (Γ + × I ) thefull subcategory in Ho(Γ + × I ) spanned by stable objects. Then the embedding Ho st (Γ + × I ) ⊂ Ho(Γ + × I ) admits a left-adjoint stabilization functorStab : Ho(Γ + × I ) → Ho st (Γ + × I ) . Proof.
Note that for any S + , S ′ + ∈ Γ + and functor X : Γ + → Top + , we havea natural map(4.8) _ s ∈ S X ( i s + ∧ id ) : _ s ∈ S X ( S ′ + ) = S + ∧ X ( S ′ + ) → X ( S + ∧ S ′ + ) , where i s : pt → S is the embedding onto s ∈ S , and the map (4.8) isfunctorial in S + , S ′ + and X . Consider the product ∆ o × Γ + × I , with theprojection π : ∆ o × Γ + × I → Γ + × I and the functor β : ∆ o × Γ + × I → Γ + × I given by the composition∆ o × Γ + × I Σ × id × id −−−−−−→ Γ + × Γ + × I m × id −−−−→ Γ + × I, where m is the product functor (3.3), and Σ : ∆ o → Γ + is the simplicialcircle. For any X ∈ Ho(Γ + × I ), let BX = π ! β ∗ X . The maps (4.8) providea map Σ ∧ π ∗ X → β ∗ X that gives rise to a map hocolim ∆ o (Σ ∧ π ∗ X ) ∼ = hocolim o ∆ Σ ∧ X ∼ = Σ X → BX, X → Ω BX , where Σ and Ω stand forthe suspension and loop space functor applies pointwise. By induction, weobtain a map Ω n B n X → Ω n +1 B n +1 X for any n ≥
1, and we can set(4.9) Stab( X ) = colim n Ω n B n X. Then Stab obviously commutes with coproducts and Kan extensions γ ! , γ : I ′ → I , and it is the main result of [S2] that if X is stable, BX is alsostable and X → Ω BX is a homotopy equivalence. Thus to show that Stabwith the natural map Id → Stab provides the required stabilization functor,it suffice to show that Stab( X ) is stable for any X ∈ Ho(Γ + × I ).To check this, we may assume that I = pt , so that X is a Γ-space. Forany such X and S + , S ′ + ∈ Γ + , denote by X ( S + , S ′ + ) the homotopy cofiber ofthe map (4.6). Say that X is n -connected , n ≥
0, if X ( S + ) is n -connected forany S + ∈ Γ + , and n -stable if it is group-like and X ( S + , S ′ + ) is n -connectedfor any S + , S ′ + ∈ Γ + . Then if a Γ-space X is m -stable, the correspondingmaps (4.6) induce isomorphisms on the homotopy groups π i for i < m , andthen for any n ≤ m , the same maps for Ω n X are isomorphisms on π i for i < m − n . Thus it suffices to prove that for any X and n ≥ B n X is2 n -stable. By induction, it further suffices to prove that BX is 0-connectedand 1-stable, and ( n + 1)-connected and ( m + 2)-stable if X is n -connectedand m -stable. For connectedness, recall that for any Y : ∆ o → Top + suchthat Y ([ q ]) is n -connected for any [ q ] ∈ ∆ o and contractible for q = 0, hocolim ∆ o Y is ( n + 1)-connected, and note that Y : ∆ o → Top + sending[ q ] ∈ ∆ o to X ( S + ∧ Σ([ q ])) satisfies this assumption for any S + ∈ Γ + . Forstability, note that BX ( S + , S ′ + ) = hocolim ∆ o δ ∗ Y, where Y : ∆ o × ∆ o → Top + sends [ q ] × [ q ′ ] to X ( S + ∧ Σ([ q ]) , S ′ + ∧ Σ([ q ′ ])),apply the isomorphism (4.5), and repeat the same argument twice. (cid:3) Remark 4.5.
For any functor γ : I ′ → I , the pullback ( id × γ ) ∗ sendsstable objects to stable objects, so that by adjunction, we have a naturalmap Stab ◦ ( id × γ ) ∗ → ( id × γ ) ∗ ◦ Stab, and the construction of Stab givenin Proposition 4.4 immediately shows that this map is an isomorphism. If I ′ = ∆ o and I = pt , then the adjoint map ( id × γ ) ! ◦ Stab → Stab ◦ ( id × γ ) ! is also an isomorphism. Indeed, in this case ( id × γ ) ∗ is fully faithful, soit suffices to check that ( id × γ ) ! sends stable objects to stable objects. Butsince hocolim ∆ o commutes with finite products, ( id × γ ) ! sends special objectsto special objects, and it obviously also preserves the group-like condition.52 emark 4.6. For any bounded category I , we have the forgetful functor e ∗ : Ho st (Γ + × I ) → Ho( I ), where(4.10) e : I × Γ + is the embedding onto pt + × I , and by virtue of Proposition 4.4, it has aleft-adjoint spectrification functor Stab ◦ e ! : Ho( I ) → Ho st (Γ + × I ). Remark 4.7.
Say that a Γ -set is a functor X : Γ + → Sets, say that itis special if X ( o ) = pt and the maps (4.6) are isomorphisms, say that itis moreover stable if the product (4.7) turns it into a group, and for anycategory I , say that X : Γ + × I → Sets is stable if so is X i : Γ + → Sets forany i ∈ I . Then a stable Γ-set is the same thing as an abelian group, so if I is bounded, we have a full embedding(4.11) Fun( I, Z ) ∼ = Fun st (Γ + × I, Sets) ⊂ Fun(Γ + × I, Sets) , where Fun st stands for the full subcategory spanned by stable objects.It is easy to see that the embedding (4.11) admits a left-adjoint stabi-lization functor Stab . If we compose it with the left Kan extension e ! ,as in Remark 4.6, then Stab ( e ! S ) ∼ = Z [ S ], the free abelian group gener-ated by the functor S : I → Sets. The functor π : Top + → Sets isleft-adjoint to the embedding Sets → Top + sending a set S to the dis-crete topological space S + , the same holds in families, and then by adjunc-tion, we have π (Stab( X )) ∼ = Stab ( π ( X )) for any X ∈ Ho(Γ + × I ) and π (Stab( e ! X )) ∼ = Z [ π ( X )] for any X ∈ Ho.
It turns out that purely by formal gameswith adjunction, Proposition 4.4 yields several useful corollaries. We willneed two of them. For the first one, say that a category I is half-additive ifit is pointed and has finite coproducts. Then for any bounded half-additivecategory I , we have a natural functor(4.12) m : Γ + × I → I sending S × i to the coproduct of copies of i numbered by elements s ∈ S ,and (4.10) is a section of (4.12). Say that X ∈ Ho( I ) is stable if so is m ∗ X ∈ Ho(Γ + × I ), and denote by Ho st ( I ) ⊂ Ho( I ) the full subcategoryspanned by stable functors. Note that since I is pointed, we have naturalmaps i ⊔ i ′ → i , i ⊔ ′ → i ′ for any i, i ′ ∈ I , and then for any X ∈ Ho st ( I ),the induced map X ( i ⊔ i ′ ) → X ( i ) × X ( i ′ ) is a weak equivalence (if i = i ′ ,53his follows from Definition 4.2, and if not, observe that i ⊔ i ′ is a retractof ( i ⊔ i ′ ) ⊔ ( i ⊔ i ′ )). For any bounded half-additive category I , the functor(4.1) extends to a functor(4.13) H o st : Ho st ( I ) → Fun( I, Ho st (Γ + )) , E
7→ H o ( m ∗ E ) , where H o ( m ∗ E ) is taken with respect to the projection Γ + × I → I . Lemma 4.8.
For any bounded half-additive category I , the full embedding Ho st ( I ) ⊂ Ho( I ) admits a left-adjoint functor Stab I : Ho( I ) → Ho st ( I ) .Proof. Take a finite set S , and consider the product Γ S + of copies of thecategory Γ + indexed by elements s ∈ S . Since Γ + is pointed, the projection τ s : Γ S + → Γ + onto the component corresponding to some s ∈ S has a left andright-adjoint functor i s : Γ + → Γ S + , and since Γ + has finite coproducts, thediagonal embedding δ S : Γ + → Γ S + has a left-adjoint functor Σ S : Γ S + → Γ + .Moreover, we have tautological anchor maps Σ S → τ s , s ∈ S , and and theseinduce a map(4.14) Σ ∗ S X → Y s ∈ S τ ∗ s X for any X ∈ Ho(Γ + ) that is an isomorphism if X is stable. But δ ∗ S is left-adjoint to Σ ∗ S , and i ∗ s is left-adjoint to τ ∗ s for any s ∈ S . Therefore byadjunction, (4.14) induces an isomorphism(4.15) Stab( δ ∗ S Y ) ∼ = Y s ∈ S Stab( i ∗ s Y )for any Y ∈ Ho(Γ S + ). In particular, if we consider the endofunctor σ S =Σ S ◦ δ S of the category Γ + , then for any Y ∈ Ho(Γ + ), we have a naturalisomorphism(4.16) Stab( σ ∗ S Y ) ∼ = Stab( Y ) S . Now take a bounded half-additive category I , with the functors (4.12) and(4.10), and let Stab I : Ho( I ) → Ho( I ) be the composition e ∗ ◦ Stab ◦ m ∗ ,with a map Id ∼ = e ∗ ◦ m ∗ → Stab I induced by the adjunction map Id → Stab.Then for any stable X ∈ Ho st ( I ), the map X → Stab I ( X ) is obviously anisomorphism, and for any Y ∈ Ho( I ), Stab I ( Y ) is stable by (4.16). (cid:3) For the second corollary, again let I be an arbitrary bounded category,take some n ≥
2, and consider the product Γ n + × I . Then it can be decom-posed as Γ + × (Γ n − × I ) in n different ways. Say that X ∈ Ho(Γ n + × I )54s polystable if it is stable with respect to all n decompositions, and letHo st (Γ n + × I ) ⊂ Ho(Γ n + × I ) be the full subcategory spanned by polystableobjects. Then smash product in Γ + induces a functor(4.17) Γ n + × I → Γ × I over I , and for any splitting Γ n + = Γ + × Γ n − and objects S + , S ′ + ∈ Γ + , S q ∈ Γ n − , we have m n (( S + ∨ S ′ + ) × S q ) ∼ = m n ( S + × S q ) ∨ m n ( S ′ + × S q ). Therefore m ∗ n sends stable objects to polystable objects and induces a functor(4.18) m ∗ n : Ho st (Γ + × I ) → Ho st (Γ n + × I ) , We then have the following result.
Lemma 4.9.
For any n ≥ and I , the functor (4.18) is an equivalence.Proof. The functor m n of (4.17) is the composition of functors m for thecategories Γ l + × I , 0 ≤ l < n , so by induction, it suffices to consider the case n = 2. Moreover, (4.18) admits a left-adjoint Stab ◦ m n ! , and we have tocheck that the adjunction maps Id → m ∗ n ◦ Stab ◦ m n ! , Stab ◦ m n ! ◦ m ∗ n → Id are invertible. By Remark 4.5, both commute with base change with respectto the category I , so it suffices to consider the case I = pt , when m is theusual smash product functor m : Γ → Γ + .Let r : [1] → Γ + be the embedding sending 0 to pt + and 1 to ∅ + ,with the unique anchor maps between them, and let t l : Γ + → Γ + × [1]be the embedding onto Γ + × l , l = 0 ,
1. Fix a splitting Γ = Γ + × Γ + ,and consider the embedding id × r : Γ + × [1] → Γ + × Γ + ∼ = Γ . Denote q = m ◦ ( id × r ) : Γ + × [1] → Γ + . Denote by Ho (Γ + × [1]) ⊂ Ho(Γ + ) and byHo st (Γ + × [1]) ⊂ Ho st (Γ + ) the full subcategories spanned by objects withcontractible t ∗ X . Then q and id × r induce functors(4.19) Ho st (Γ + ) q ∗ −−−−→ Ho st (Γ + × [1]) ( id × r ) ∗ ←−−−−− Ho st (Γ ) , and it suffices to prove that both are equivalences.For q ∗ , for any S + ∈ Γ + , consider the functor ε ( S + ) : V o → Γ + × [1]sending 0, 1, o to S + × S + × ∅ + ×
1, and note that q o ◦ ε ( S + ) o : V → Γ o + sends 0 to S + and the rest to the initial object pt + ∈ Γ o + , so that it isnaturally augmented by S + . Moreover, this defines a framing for q o in thesense of Lemma 1.14, and if we compute the homotopy right Kan extension q ∗ using this framing, we see that q ∗ X for any X ∈ Ho(Γ + × [1]) fits into a55omotopy cartesian square(4.20) q ∗ X −−−−→ t ∗ X y y X ( ∅ + × −−−−→ t ∗ X. Then if X is in Ho st (Γ + × [1]), the bottom arrow in (4.20) is a homotopyequivalence, so that q ∗ X ∼ = t ∗ X is stable, and q ∗ then gives an equivalenceinverse to q ∗ .For ( id × r ∗ ), let h a, s i be the anchor/structural factorization system onΓ + . Then r : [1] → (Γ + ) a is full, so that by Example 1.6, we have aframing for r given by categories [1] / rs S + , S + ∈ Γ + . Explicitly, we have[1] / rs S + ∼ = S < , the discrete category S with the added initial object o . Thisinduces a framing for id × r , and then for any X ∈ Ho (Γ + × [1]) and S + ∈ Γ + ,we have(4.21) ( id × r ) ! X | Γ + × S + ∼ = _ s ∈ S t ∗ X ∼ = t ∗ X ∧ S + . But for any Γ-space Z ∈ Ho(Γ + ), the product Z ∧ S + = Z ∨ · · · ∨ Z is of theform δ ∗ Z ∨ S for the object Z ∨ S = _ s ∈ S τ ∗ s Z ∈ Ho(Γ S + ) , and then as in the proof of Lemma 4.8, the isomorphism (4.15) shows thatwe have(4.22) Stab( Z ∧ S + ) ∼ = Stab( Z ) S . Therefore if X is stable, (Stab ◦ ( id × r ) ! X ) | Γ + × S + ∼ = ( t ∗ X ) S by (4.21). Thenfirstly, Stab(( id × r ) ! X ) that a priori only lies in Ho st (Γ + × Γ + ) is polystable,secondly, the adjunction map X → ( id × r ) ∗ ◦ Stab ◦ ( id × r ) ! X is an isomor-phism, and, last but not least, the adjunction map Stab(( id × r ) ! r ∗ Y ) → Y is an isomorphism for any polystable Y ∈ Ho st (Γ ). Then Stab ◦ ( id × r ) ! isthe inverse equivalence to ( id × r ) ∗ , and we are done. (cid:3) A convenient way to package allthe data associated to a symmetric monoidal category is the same Segal ma-chine as in Definition 4.2, with functors to spaces replaced by Grothendieckcofibrations. 56 efinition 4.10. A unital symmetric monoidal structure on a category C is a cofibration B ∞ C → Γ + equipped with an equivalence B ∞ C pt + ∼ = C suchthat B ∞ C o is the point category pt , and for any S, S ′ ∈ Γ, the functor a ! × a ′ ! : B ∞ C S + ∨ S ′ + → B ∞ C S + × B ∞ C S ′ + induced by the maps (3.2) is an equivalence of categories.Informally, if we consider the set { , } with two elements, then the twoanchor maps { , } + → pt + identify the fiber B ∞ C { , } + with the product C ,and the unique structural map { , } + → pt + provides the product functor µ : C → C . All the associativity, commutativity and unitality constraintsare packaged into the maps (2.2) for the cofibration B ∞ C → Γ + . Example 4.11.
For any bounded category I , and any category C equippedwith a unital symmetric monoidal structure B ∞ C , the functor categoryFun( I, C ) carries a natural pointwise unital symmetric monoidal structuregiven by B ∞ Fun( I, C ) ∼ = Fun( I, B ∞ C / Γ + ). Example 4.12.
A category C that has finite cartesian products carries aunital symmetric monoidal structure B ∞ C given by these products. For-mally, take the embedding e : pt → Γ + of (4.10), observe that the fibra-tion B ′∞ C = ( e o ∗ C o ) o → Γ + is also a cofibration, and take the preimage B ∞ C = ζ ( o ) − (1) ⊂ B ′∞ C of the terminal object 1 ∈ C = ( B ′∞ C ) o underthe functor (2.1) for the terminal object o = ∅ + ∈ Γ + . By Lemma 2.11 (ii), B ∞ C → Γ + is a cofibration, and the embedding B ∞ C ⊂ B ′∞ C has a left-adjoint functor λ cocartesian over Γ + . Example 4.13.
A commutative monoid X considered as a discrete cate-gory carries a unital symmetric monoidal structures B ∞ X = Γ + [ X ∞ ], where X ∞ : Γ + → Sets + is the discrete special Γ-space corresponding to X , withthe unit element as the distinguished point. Definition 4.14. A lax monoidal structure on a functor F : C ′ → C betweentwo categories C , C ′ equipped with unital symmetric monoidal structures B ∞ C resp. B ∞ C ′ is a functor B ∞ F : B ∞ C ′ → B ∞ C over Γ + , cocartesianalong anchor maps in Γ + and equipped with an isomorphism B ∞ F pt + ∼ = F .A lax monoidal structure B ∞ F is monoidal if it is cocartesian over all maps.The essential structure for a lax monoidal functor F is the map µ ◦ F → F ◦ µ ′ provided by (2.7) for the functor B ∞ F . The functor F is monoidaliff this map is an isomorphism. 57 xample 4.15. Assume that a functor γ : C → C ′ between two symmetricmonoidal categories is equipped with a symmetric monoidal structure B ∞ γ ,and at the same time admits a right-adjoint γ † : C ′ → C . Then B ∞ γ admits aright-adjoint B ∞ γ † over Γ + that defines a lax symmetric monoidal structureon γ † . Example 4.16.
For any category C equipped with a unital symmetric mo-noidal structure B ∞ C , the opposite category C o carries a unital symmetricmonoidal structure B ∞ C o = B ∞ C ⊥ o . For any two categories C , C ′ withunital symmetric monoidal structures B ∞ C , B ∞ C ′ , the product C ×C ′ carriesa unital symmetric monoidal structure B ∞ ( C × C ′ ) = B ∞ C × Γ + B ∞ C ′ . Forany unital symmetric monoidal category C , the Yoneda pairing(4.23) Y : C o × C → Setscarries a natural lax monoidal structure B ∞ Y with respect to the cartesianproduct in Sets. Example 4.17.
For any object A ∈ C in a unital symmetric monoidal cate-gory C , giving a lax monoidal structure B ∞ i A on the embedding i A : pt → C is equivalent to turning A ∈ C into a unital commutative associative algebra. Example 4.18.
By Example 4.12 and Example 4.16, any category I thathas finite coproducts is a symmetric unital monoidal category with respectto coproducts. In particular, this applies to Γ; the corresponding category B ∞ Γ is ar s (Γ + ), where s is the class of structural maps. For the oppo-site category Γ o , we have B ∞ Γ o ∼ = tw s (Γ + ). The structural cofibration B ∞ Γ ∼ = ar s (Γ + ) → Γ + admits a right-adjoint η : Γ + → B ∞ Γ, and by Ex-ample 4.15 and Example 4.17, this turns the one-point set pt ∈ Γ into analgebra object in C . This algebra object is universal: for any unital symmet-ric monoidal category C and object A ∈ C , a lax monoidal structure B ∞ i A on the embedding i A : pt → C factors uniquely as B ∞ i A ∼ = B ∞ I A ◦ η for aunique monoidal functor I A : Γ → C . Explicitly, B ∞ I A = η Γ + ! B ∞ i A , wherethe left Kan extension is given by (2.6). Example 4.19.
The isomorphism groupoid I of a unital symmetric monoi-dal category I carries a unital symmetric monoidal structure B ∞ I = ( B ∞ I ) ♮ .For Γ with B ∞ Γ of Example 4.18, the groupoid Γ also has a universal prop-erty: for any object c ∈ C in a unital symmetric monoidal category, thereexists a unique symmetric monoidal functor γ : Γ → C sending pt to c . Inparticular, if we take the monoid N of non-negative integers with respect to58ddition, and treat is a discrete unital symmetric monoidal category as inExample 4.13, then 1 ∈ N defines a monoidal functor c : Γ → N sending aset to its cardinality. Note that B ∞ c has no sections.For any unital symmetric monoidal categories I , C with bounded I , laxmonoidal functors from I to C form a category that we denote Fun ∞ ( I, C ).More generally, if we are given a functor γ : C → I equipped with amonoidal structure B ∞ γ , then sections of B ∞ γ that are lax monoidal func-tors form a category that we denote Sec ∞ ( I, C ). We tautologically haveFun ∞ ( I, C ) ∼ = Sec ∞ ( I, I × C ), and sections of the tautological projection
C → pt are algebra objects in C , as in Example 4.17. It is useful to observethat under favourable conditions, arbitrary lax monoidal functors can be alsodescribed as algebra objects. Namely, informally, for any small category I and symmetric monoidal category C with a product − ⊗ − , the product in C induces a box product(4.24) − ⊠ − : Fun( I, C ) → Fun( I , C ) , ( F ⊠ F ′ )( i × i ′ ) = E ( i ) ⊗ E ( i ′ )If I itself is unital symmetric monoidal, we also have the product functor m : I → I and the pullback functor m ∗ : Fun( I, C ) → Fun( I , C ). It mighthappen that m ∗ has a left-adjoint m ! that commutes with all products − ⊗ c , c ∈ C . In this case, we can turn Fun( I, C ) into a symmetric monoidal tensorcategory by considering the convolution product(4.25) F ◦ F ′ = m ! ( F ⊠ F ′ ) . More formally, let ( a, s ) be the anchor/structural factorization system onΓ + , and use the technology of Subsection 2.4. If denote by B s ∞ I → Γthe restriction the cofibration B ∞ I → Γ + to Γ = (Γ + ) s ⊂ Γ + , then itimmediately follows from (2.45) that the cofibration(4.26) B ∞ Fun( B s ∞ I/ Γ , C ) ∼ = F un a ( B ∞ I/ Γ + , B ∞ C / Γ + ) → Γ + defines a symmetric monoidal structure on the category Fun( B s ∞ I/ Γ , C ) of(2.31). Explicitly, objects in Fun(( B ∞ I ) s / Γ , C ) are pairs h S, F i , S ∈ Γ, F : I S → C , and symmetric monoidal structure is induced by the boxproduct (4.24): we have h S, F i ◦ h S ′ , F ′ i = h S ⊔ S ′ , F ⊠ F ′ i . The projectionFun( B s ∞ I/ Γ , C ) → Γ o is monoidal with respect to the monoidal structure B ∞ Γ o ∼ = tw s (Γ + ) of Example 4.18, and (2.43) reads as(4.27) Fun ∞ ( I, C ) ∼ = Sec ♮ ∞ (Γ o , Fun( B s ∞ I/ Γ , C ) , ♮ ∞ ( − , − ) stands for the full subcategory spanned by cocartesiansections. We also have the reflection of the cofibration (4.26) over B ∞ Γ o ,and one immediately checks by induction that if m ∗ admits a left-adjoint m ! such that m ! ( a ⊗ c ) ∼ = m ! a ⊗ c , a ∈ Fun( I , C ), c ∈ C , then (2.46) satisfiesthe conditions of Lemma 2.19, so that (2.47) is a cofibration. Then again,one checks by (2.45) that(4.28) B ∞ Fun( I, C ) ∼ = Fun a (( B ∞ I/ Γ + , B ∞ C / Γ + ) → Γ + defines a symmetric monoidal structure on Fun( I, C ). This is our convolutionmonoidal structure, and (2.49) then provides an equivalence(4.29) Fun ∞ ( I, C ) ∼ = Sec ∞ ( pt , Fun( I, C )) , so that lax monoidal functors from I to C are identified with algebra objectsin Fun( I, C ). Remark 4.20.
A monoidal functor γ : I ′ → I between bounded symmetricmonoidal categories induces a cocartesian functor B s ∞ γ : B s ∞ I ′ → B s ∞ I ,and the pullback functor ( B s ∞ γ ) ∗ is naturally monoidal with respect to thestructures (4.26). If the cofibrations (4.28) exist, then γ ∗ : Fun( I, C ) → Fun( I ′ , C ) is naturally lax monoidal (but only lax since γ ∗ commutes with m ∗ but not necessarily with m ! ). However, if γ ∗ admits a left-adjoint Kanextension functor γ ! , then γ ! does commute with m ! . In this case, B ∞ γ ∗ admits a left-adjoint B ∞ γ ! over Γ + that is cocartesian and defines a monoidalstructure on γ ! . Example 4.21.
Let C be a cocomplete unital symmetric monoidal categorysuch that c ⊗ − : C → C preserves colimits for any c ∈ C , and let N , Γ bethe unital symmetric monoidal categories of Example 4.19. Then Fun( N , C )is the monoidal category of non-negatively graded objects in C . Objectsin Fun(Γ , C ) are “symmetric sequences” of [HSS]. If c : Γ → N is thecardinality functor, then the pullback c ∗ is fully faithful, and the left Kanextension c ! : Fun(Γ , C ) → Fun( N , C ) is monoidal by Remark 4.20, so thatFun( N , C ) is a monoidal localization of Fun(Γ , C ). There are other interestinglocalizations. For example, if C = Z -mod is the category of abelian groups,then we have the symmetric sequence Z q ∈ Fun(Γ , C ) given by the signrepresentations, the pointwise tensor product Z q ⊗ Z q is Z , and we have a fullembedding c ∗ q : Fun( N , Z ) → Fun(Γ , Z ) given by c ∗ q ( E ) = c ∗ E ⊗ Z q . It has aleft-adjoint c q ! sending E q ∈ Fun(Γ , Z ) to c ! ( E q ⊗ Z q ), and by Lemma 2.11 (ii),this induces a new symmetric monoidal product on Fun( N , Z ) such that c q !
60s monoidal. This new product is the old one but twisted by the usualhomological signs, so that commutative algebras in Fun( N , Z ) with the newsymmetric monoidal structure are graded-commutative algebras over Z . Γ -spaces. We now recall the lesser-known part of[S2] that deals with multiplications. We note that the smash product de-fines a unital symmetric monoidal structure both on Γ + and on Top + . Sincesmash product preserves weak equivalences, it also induces a unital symmet-ric monoidal structure on Ho. Definition 4.22. A multiplicative Γ -space is a lax monoidal functor X fromΓ + to Top + .By virtue of (4.29), multiplicative Γ-spaces correspond to algebra ob-jects in Fun(Γ + , Top + ) with respect to the convolution product, but thisnot very useful since the convolution product is not homotopy invariant. Toconstruct a homotopy invariant formalism, it is better to use (4.27). For anybounded unital symmetric monoidal category I , Fun( B s ∞ I/ Γ , Top + ) → Γ o isa cofibration with fibers Fun( I n , Top + ), n ≥
0, and its transition functors re-spect weak equivalences. Therefore it induces a cofibration Ho( B s ∞ I/ Γ) withfibers Ho( I n ), and since smash product in Top + also preserves weak equiv-alences, (4.26) induces a symmetric monoidal structure B ∞ Ho( B s ∞ I/ Γ) onHo( B s ∞ I/ Γ). We still have a cofibration B ∞ Ho( B s ∞ I/ Γ) → B ∞ Γ o , andsince homotopy left Kan extensions commute with smash product − ∧ X for any X ∈ Top + , the reflection ( B ∞ Ho( B s ∞ I/ Γ) /B ∞ Γ o ) ⊳ again satisfiesthe conditions of Lemma 2.19. As in (2.47), we can then restrict it to thesubcategory Γ + ⊂ ar s (Γ + ) = B ∞ Γ, and obtain a cofibration(4.30) B ∞ Ho( I ) → Γ + that defines a convolution symmetric monoidal structure on the homotopycategory Ho( I ). We still have the equivalence(4.31) Sec ♮ ∞ (Γ o , Ho( B s ∞ I/ Γ)) ∼ = Sec ∞ ( pt , Ho( I )) , and we can treat an object in either of these two equivalent categories asan enhanced version of a lax monoidal functor from I to Ho. By (4.27),a geniune lax monoidal functor I → Top + generates such an object bylocalization, and forgetting the enhancement corresponds to considering theassociated functor (4.1) that is lax symmetric monoidal with respect tothe monoidal structure (4.30). We also note that as in Remark 4.20, the61omotopy Kan extension γ ! with respect to a monoidal functor γ : I ′ → I carries a natural monoidal structure.Now take a bounded unital symmetric monoidal category I , and con-sider the product Γ + × I with the product monoidal structure B ∞ (Γ + × I )of Example 4.16. Then Ho(Γ + × I ) carries a unital symmetric monoidalstructure B ∞ Ho(Γ + × I ) of (4.30). Proposition 4.23.
Let B ∞ Ho st (Γ + × I ) ⊂ B ∞ Ho(Γ + × I ) be the full sub-category spanned by Ho st (Γ + × I ) S ⊂ Ho(Γ + × I ) S ∼ = B ∞ Ho(Γ + × I ) S + , S ∈ Γ . Then the induced projection B ∞ Ho st (Γ + × I ) → Γ + is a cofibrationthat defines a unital symmetric monoidal structure on Ho st (Γ + × I ) , and thestabilization functor Stab of Proposition 4.4 extends to a monoidal functor B ∞ Stab : B ∞ Ho(Γ + × I ) → B ∞ Ho st (Γ + × I ) .Proof. For any bounded category I , say that a map f in the categoryHo(Γ + × I ) is stably trivial if Stab( f ) is invertible. Moreover, for any S ∈ Γ,say that a map g in Ho(Γ + × I ) S is stably trivial if so is each of its components g s , s ∈ S . Then by Lemma 2.11 (ii), it suffices to prove that the transitionfunctors f ! of the cofibration (4.30) send stably trivial maps to stably trivialmaps.If f is an anchor map, then f ! commutes with stabilization by definition,so by induction, it suffices to consider the case when f : { , } + → pt + is the structural map that induces the product on Ho(Γ + × I ). Explicitly,if we denote by m : (Γ + × I ) → Γ + × I the smash product on Γ + × I ,then we have to check that for any stably trivial map g in Ho(Γ + × I )and any X ∈ Ho(Γ + × I ), m ! ( g ⊠ X ) is stably trivial. Moreover, since m ∗ sends stable objects to stable objects, and the stabilization functor ofProposition 4.4 commutes with base change with respect to I , this amountsto checking that for any Y ∈ Ho, g ∧ Y is stably trivial, and it suffices tocheck it for I = pt .Indeed, if Y = S + is a finite pointed set, then the claim immediatelyfollows from (4.22). But by adjunction, a homotopy colimit of stably trivialmaps is stably trivial. Therefore the claim is clear for arbitrary pointed sets,and then for homotopy colimits over ∆ o of arbitrary simplicial pointed sets.This is the whole Ho. (cid:3) Remark 4.24.
The category Ho st (Γ + ) is of course just the category of con-nective spectra, the connective part of the standard t -structure on the stablehomotopy category, and the monoidal structure of Proposition 4.23 is thestandard smash product. A commutative associative unital algebra object62n Ho st (Γ + ) is then what used to be called a “commutative associative uni-tal connective ring spectrum” at the time of [S2], and Proposition 4.23 alsoshows that any multiplicative Γ-space generates such a thing by stabiliza-tion. The modern notion of an E ∞ -ring spectrum is of course much stronger;however, the old naive version will be sufficient for our purposes. Remark 4.25.
Even if X is a commutative ring spectrum in the weak senseof Remark 4.24, the homotopy groups π q ( X ) still form a graded commuta-tive Z -algebra. To see this in the stabilization formalism, one first observesthat by the same argument as in Proposition 4.23, the stabilization func-tor Stab of Remark 4.7 creates a unital symmetric monoidal structure onFun st (Γ + × I ), and the equivalence (4.11) sends it to the convolution producton Fun( I, Z ) (in particular, if I = pt , we recover the standard tensor productof abelian groups). Now take the circle S ∈ Ho, extend it to a monoidalfunctor Γ → Ho, pt S as in Example 4.19, and let Σ q : Γ o ∼ = Γ → Ho o be the opposite functor. Then any algebra object X in Ho st (Γ + ) defines alax monoidal functor Ho( X ) : Γ + → Ho, and we can consider the Yonedapairing π q ( X ) = Y (Σ q × Ho( X )) : Γ × Γ + → Setsof (4.23). This is a lax monoidal functor, and moreover, since X is stable, π q ( X ) is stable as an object in Fun st (Γ + × Γ). Thus by (4.11), π q ( X ) isa symmetric sequence of abelian groups that is an algebra with respect tothe convolution product of Example 4.21. One then checks that it actuallylies in the essential image of the full embedding c ∗ q , thus gives a gradedcommutative algebra over Z .Assume now given a half-additive category I , and assume that I isequipped with a unital symmetric monoidal structure. Then we say thatthe structure is distributive if 0 ⊗ i = 0 for any i ∈ I , where 0 ∈ I is theinitial object, and the natural map ( i ⊗ i ) ⊔ ( i ′ ⊗ i ) → ( i ⊔ i ′ ) ⊗ i is anisomorphism for any i , i ′ , i ∈ I . In this case, the functors (4.10) and (4.12)are monoidal, and we have the following corollary of Proposition 4.23. Corollary 4.26.
Assume given a bounded half-additive category I , and as-sume that it is equipped with a distributive unital symmetric monoidal struc-ture. As in Proposition 4.23, let B ∞ Ho st ( I ) ⊂ B ∞ Ho( I ) be the full sub-category spanned by Ho st ( I ) S ⊂ Ho( I ) S ∼ = B ∞ Ho( I ) S + , S ∈ Γ . Then theinduced projection B ∞ Ho st ( I ) → Γ + is a cofibration that defines a uni-tal symmetric monoidal structure on Ho st ( I ) , and the stabilization functor Stab I of Lemma 4.8 carries a monoidal structure B ∞ Stab I . roof. By Lemma 2.11 (iii), it suffices to construct the left-adjoint functor B ∞ Stab I . By Lemma 4.8, B ∞ Stab I = B ∞ t ∗ ◦ B ∞ Stab ◦ B ∞ m ∗ works. (cid:3) Let us now describe a linear version of the stabiliza-tion formalism. Assume given a commutative ring k . For any bounded cate-gory I , denote by Ho( I, k ) the localization of the category Fun( I, ∆ o k -mod)with respect to weak equivalences. By Dold-Kan, ∆ o k -mod is equivalent tothe category C ≥ ( k ) of chain complexes of k -modules bounded from belowby 0 (in homological degrees), weak equivalences are quasiisomorphisms,and Ho( I, k ) is the connective part of the standard t -structure on the de-rived category D ( I, k ) of Subsection 1.4, while (4.1) is induced by (1.16). Afunctor γ : I ′ → I gives rise to a pullback functor γ ∗ : Ho( I, k ) → Ho(
I, k ),and we have the left and right homotopy Kan extensions γ ! , γ ∗ (where inhomological terms, γ ! is the left-derived functor of the usual left Kan exten-sion, and γ ∗ is the right-derived functor of the right Kan extension composedwith truncation with respect to the standard t -structure). The isomorphisms(1.14), (1.13), (2.11) and Lemma 1.14 also hold for the categories Ho( − , k ),and a weak equivalence γ : I ′ → I induces an equivalence of categories γ ∗∀ : Ho ∀ ( I, k ) ∼ = Ho ∀ ( I ′ , k ).Since every k -module has a flat resolution, Ho( I, k ) can also be obtainedby localizing the category Fun( I, ∆ o k -mod fl ), where k -mod fl ⊂ k -mod is thefull subcategory spanned by flat k -modules. Then ∆ o k -mod fl is symmetricmonoidal, the product respects weak equivalences, and the same procedureas in Subsection 4.5 provides a unital symmetric monoidal structure onHo(Γ + , k ). We have the forgetul functor(4.32) ϕ : k -mod fl → Sets + sending a k -module M to its underlying set with 0 ∈ M as the distinguishedpoint, and it has a left-adjoint reduced span functor span k : Sets + → k -mod fl sending a pointed set S + to the quotient k [ S + ] /k · o of the free k -module k [ S + ] generated by S + by the submodule k · o spanned by the distinguishedpoint o ∈ S + . The functor span k is monoidal with respect to the smashproduct on Sets + , so that (4.32) is lax monoidal by adjunction, as in Ex-ample 4.15. On the level of homotopy categories, for any bounded I , (4.32)induces a functor ϕ : Ho( I, k ) → Ho( I ) sending a simplicial k -module to thegeometric realization of the underlying simplicial set, with its left-adjoint span k : Ho( I, k ) → Ho( I ), and these functors commute with pullbacks γ ∗ with respect to functors γ : I ′ → I . 64ll the results about stabilization also have counterparts for the k -linearhomotopy categories Ho( − , k ). Namely, we say that X ∈ Ho(Γ + × I, k ) isstable if so is ϕ ( X ) ∈ Ho(Γ + × I, k ) and analogously, if I is bounded andhalf-additive, X ∈ Ho(
I, k ) is stable if so if ϕ ( X ). We denote by(4.33) Ho st (Γ + × I, k ) ⊂ Ho(Γ + × I, k ) , Ho st ( I, k ) ⊂ Ho(
I, k )the full subcategories spanned by stable objects, and the stabilization func-tors of Proposition 4.4 resp. Lemma 4.8 also provide stabilization functorsStab, Stab I left-adjoint to the embeddings (4.33) (one has to replace Top + with ∆ o k -mod, and use the truncated homological shift instead of the loopfunctor Ω). These stablization functors are monoidal with respect to naturalsymmetric monoidal structures. We also have natural isomorphisms(4.34) ϕ ◦ Stab ∼ = Stab ◦ ϕ, ϕ ◦ Stab I ∼ = Stab I ◦ ϕ, where ϕ are the forgetful functors induces by (4.32). In particular, the func-tor Stab of Remark 4.7 is given by Stab ( X ) = τ ≤ (Stab( Z [ X ])), where τ ≤ is the truncation with respect to the standard t -structure. A new fea-ture in the k -linear case is that finite products in ∆ o k -mod coincide withcoproducts, so that not only pullbacks but also left Kan extensions preservestability: for any bounded functor γ : I ′ → I bewteen bounded categories,( id × γ ) ∗ and ( id × γ ) ! induce an adjoint pair of functors(4.35) ( id × γ ) ∗ : Ho st (Γ + × I, k ) → Ho st (Γ + × I ′ , k ) , ( id × γ ) ! : Ho st (Γ + × I ′ , k ) → Ho st (Γ + × I, k ) , and similarly for Ho st ( I, k ), Ho st ( I ′ , k ) for a half-additive I . Then k -linearstabilization commutes both with pullbacks and left Kan extensions, so that(4.34) and the isomorphism Stab ◦ ( id × γ ) ∗ ∼ = ( id × γ ) ∗ ◦ Stab of Remark 4.5induce by adjunction a natural isomorphism(4.36) Stab ◦ ( id × γ ) ! ◦ ϕ ∼ = ( id × γ ! ) ◦ ϕ ◦ Stab , and similarly for Stab I , Stab I ′ . Remark 4.27.
For any Γ-space X ∈ Ho(Γ + ) and any commutative ring k , the isomorphism ϕ ◦ Stab ∼ = Stab ◦ ϕ induces a map Stab( span k ( X )) → span k (Stab( X )), However, this map is not an isomorphism. In effect, forany X ∈ Ho, span k ( X ) ∈ Ho( k ) is represented by the homology complexof the space X with coefficients in k , so that span k (Stab( X ))( pt + ) is thehomology of the infinite loop space Stab( X )( pt + ). On the other hand,Stab( span k ( X ))( pt + ) is the homology of the corresponding spectrum (thisis more-or-less obvious from the construction, or see a proof in [Ka2]).65lternatively, one can construct the stabilization functors directly bypurely homological means; we only give a sketch, and refer the reader to[Ka1, Section 3] for details. We note that the group-like requirement inthe definition of stability is automatic in the k -linear case, and again, theproducts and coproducts in Ho( k ) coincide. Thus in the k -linear context,“stable” just means “additive”, or rather, “half-additive” — that is, sendingcoproducts to coproducts. The category Ho st (Γ + , k ) is then equivalent toHo( k ), with the equivalence Ho( k ) → Ho st (Γ + , k ) ⊂ Ho(Γ + , k ) sending M ∈ Ho( k ) to M ⊗ k T , where T ∈ Ho(Γ + , k ) is the reduced span functor, S + span k ( S + ). Therefore the left-adjoint functor Stab : Ho(Γ + , k ) → Ho( k ) isgiven by(4.37) Stab( M ) = M L ⊗ Γ + T o , T o ( S + ) = Hom k ( T ( S + ) , k ) , where L ⊗ Γ o + is the derived tensor product (1.25) over the small category Γ + .To compute it, one needs to choose a projective resolution of T o in theabelian category Fun(Γ o + , k ). This is greatly simplified by first constructingan equivalence(4.38) Fun(Γ o + , k ) ∼ = Fun(Γ o − , k ) , where Γ − is the category of finite sets and surjective maps between them,and doing it in such a way that T o is sent to t ∈ Fun(Γ o − , k ) with t ( S ) = k for S = pt and 0 otherwise. Then the functors s n ∈ Fun(Γ o − , k ) representedby sets of cardinality n , n ≥ o − , k ), with s = k being the constant functor with value k .We have an exact sequence s η −−−−→ s −−−−→ t −−−−→ , and it extends to a resolution P q of t given by(4.39) P n = s ⊗ n , d = X ≤ i ≤ n id i − ⊗ η ⊗ id n − i on P n , where one needs to check separately that s ⊗ n is projective for every n . Asan additional bonus, the equivalence (4.38) sends the convolution monoidalstructure on Fun(Γ + , k ) to the pointwise monoidal structure on Fun(Γ o − , k ),so that the resolution (4.39) is monoidal, and also describes the monoidalstructure on Stab. 66 emark 4.28. Another advantage of the resolution (4.39) is that each term P n is the sum of a finite number of representable objects; this implies thatadditivization commutes with arbitrary products. More generally, (4.37)makes sense for any M ∈ D (Γ + , k ), and Stab commutes with arbitraryproducts in D ≤ n (Γ + , k ) for any fixed n . Remark 4.29.
Although unknown to the author at the time, all the con-structions of [Ka1, Section 3] were actually discovered much earlier by T.Pirashvili. The general additivization construction is summarized nicelyin [P1], while (4.38) is in [P2] (and should be called “Pirashvili-Dold-Kanequivalence”).
To illustrate stabilization and additivization, wetake a commutative ring k and consider functors F from k -mod fl to it-self. The most basic example is F = span k ◦ ϕ : k -mod fl → k -mod fl , thecomposition of the forgetful functor (4.32) and its left-adjoint. By Re-mark 4.27, Stab( F ) sends a flat k -module M to Q ( M/k ) = Stab( F )( M )in Ho( k ) ∼ = Ho st ( k ) that represents its stable homology with coefficients in k (that is, the homology of the corresponding Eilenberg-Mac Lane spectrum).Since F is symmetric monoidal, Stab( F ) is also symmetric monoidal, so thatfor any commutative flat k -algebra A , Q ( A/k ) is a commutative ring objectin Ho( k ). If k = F p is a prime field, then the homology algebra H q ( Q ( k/k ))is the dual Steenrod algebra. We remind the reader that if p is odd, this isthe free graded-commutative algebra(5.1) H q ( Q ( k/k )) = k [ β, ξ i , τ i ]generated by the Bokstein element β of degree 1, and elements τ i , ξ i , i ≥ ξ i = 2 p i −
2, deg τ i = 2 p i − p = 2, then the formulais the same but without τ i , deg ξ i = 2 i +1 −
1, and one has to remember thatin char
2, “graded-commutative” means “commutative”.If one computes Stab( F ) by (4.37) with the resolution (4.39), one canlift it to a functor k -mod fl → ∆ o k -mod ∼ = C ≥ ( k ), M Q q ( M/k ). If k = Z , this functor actually coincides on the nose with the cubical con-struction Q q ( M ) of Eilenberg and Mac Lane. It is lax monoidal (althoughnot symmetric monoidal, since (4.39) only has a non-commutative algebrastructure), so that for any k -algebra A , we have a DG algebra Q q ( A/k ).67eneralizing this further, for any additive flat k -linear category A , onecan define a k -linear DG category Q q ( A/k ) with the same objects, and withmorphism complexes given by Q q ( A/k )( a, a ′ ) = Q q ( A ( a, a ′ ) /k ). Moreover,one can extend Q q ( − /k ) to a functor ∆ o k -mod fl → C ≥ ( k ) by applyingit pointwise to a simplicial group, and then totalizing the bicomplex ob-tained by the Dold-Kan equivalence from the resulting object Q q ( M/k ) ∈ ∆ o C ≥ ( k ). Then the extended functor Q q can be applied pointwise to sim-plicial objects in Fun( A, k ) pointwise-flat over k , and this provides a functor Q : Ho( A, k ) → D ( Q q ( A/k )) to the derived category of modules over theDG category Q q ( A/k ). By definition, it factors through the stabilizationfunctor Stab, and in fact induces an equivalence(5.2) Q : Ho st ( A, k ) ∼ = D ≥ ( Q q ( A/k )) , where D ≥ ( Q q ( A/k )) ⊂ D ( Q q ( A/k )) is spanned by modules concentrated innon-negative homological degrees. To recover the whole triangulated cate-gory D ( Q q ( A/k )), take the closure of Ho st ( A, k ) ⊂ Ho(
A, k ) = D ≤ ( A, k ) ⊂D ( A, k ) with respect to homological shifts.For other examples, consider polynomial functors k -mod fl → k -mod.The simplest such is the tensor power functor T n ( M ) = M ⊗ n for somefixed n , but this is not interesting: (4.15) immediately implies that we haveStab( T n ) = 0 as soon as n ≥
2. For an interesting example, fix a prime p ,and consider the cyclic power functor C given by(5.3) C ( M ) = H ( Z /p Z , M ⊗ k p ) = τ ≤ C q ( Z /p Z , M ⊗ k p ) , where the Z /p Z -action on M ⊗ p is generated by the longest cycle permuta-tion, and C q ( Z /p Z , − ) ∈ D ( k ) stands for the cohomology complex of thegroup Z /p Z . Denote R = Stab( C ). Lemma 5.1.
For any M ∈ k -mod fl , we have a functorial isomorphism (5.4) R ( M ) ∼ = τ ≤ ˇ C q ( Z /p Z , M ⊗ k p ) , where ˇ C q ( Z /p Z , − ) ∈ D ( k ) stands for the Tate cohomology complex, and thecomposition of (5.4) with the natural map C ( M ) → R ( M ) is induced by theembedding C q ( Z /p Z , M ⊗ k p ) → ˇ C q ( Z /p Z , M ⊗ k p ) .Proof. To compute the target of the hypothetical isomorphism (5.4), choosea resolution P q of the trivial k [ Z /p Z ]-module k by free k [ Z /p Z ]-modules P i =68 i ⊗ k k [ Z /p Z ], V i ∈ k -mod, let P ′ q be the shifted cone of the augmentationmap P q → k , and let R q ( M ) = H ( Z /p Z , P ′ q ⊗ k M ⊗ k p ) . Then R q ( M ) ∈ C ≥ ( k ) represents the target of (5.4), and the embedding k → P ′ q induces a functorial map(5.5) C ( M ) → R q ( M ) . For any i ≥
0, we have R i +1 ( M ) ∼ = V i ⊗ k M ⊗ p , and since Stab( T p ) = 0,we also have Stab( R i +1 ) = 0, so that the map in Ho( k -mod , k ) repre-sented by (5.5) is stably trivial. On the other hand, for any two flat k -modules M , M ∈ k -mod fl , the quotient ( M ⊕ M ) ⊗ k p / ( M ⊗ k p ⊕ M ⊗ k p ) isa free k [ Z /p Z ]-module with trivial Tate cohomology, so that the object inHo( k -mod , k ) given by R q ( M ) is stable. Stabilizing (5.5), we get (5.4). (cid:3) Up to now, all our example were k -linear. However, since stabilizationcommutes with the forgetul functor (4.32), even a non-linear map F → F between functors F , F : k -mod → k -mod induces a map between theirstabilizations. For an example of this, assume that k is a perfect field ofcharacteristic p , and for any k -vector space M ∈ k -mod, consider the map M → M ⊗ p sending m ∈ M to m ⊗ p . This map is Z /p Z -equivariant andfunctorial, thus induces a map(5.6) ψ : ϕ = ϕ ◦ Id → ϕ ◦ C, and this gives rise to a functorial map(5.7) Stab( ψ ) : ϕ ( M ) → ϕ ( R ( M ))in Ho st (Γ + ) for any k -vector space M ∈ k -mod. Lemma 5.1 easily impliesthat Stab( ψ ) is an isomorphism on π , but ϕ ( R ( M )) also has higher homo-topy groups. In fact, (5.4) induces a functorial exact triangle(5.8) C ( M ) −−−−→ R ( M ) a −−−−→ C q ( Z /p Z , M ⊗ p )[1] −−−−→ in D ≥ ( k ), where C q ( Z /p Z , − ) stands for the group homology complex, andcomposing the projection a with (5.7), we obtain a map(5.9) ψ ′ = a ◦ ψ : ϕ ( M ) → ϕ ( C q ( Z /p Z , M ⊗ p )[1]) . Both the source and the target of ψ ′ are k -linear, the source is discrete, andthe target is 1-connective, so were ψ ′ to be k -linear, it would vanish. It doesnot, even for the one-dimensional space M = k . In principle, the whole map(5.9) can be computed explicitly in terms of the Steenrod power operations,see [NS], but for our purposes, the following it sufficient.69 emma 5.2. The map ψ : ϕ ( k ) → ϕ ( k [1]) obtained by composing (5.9) with the projection C q ( Z /p Z , k ) → H ( Z /p Z , k ) = k is the composition ofthe Frobenius endomorphism k → k and the Bokstein map.Proof. More generally, for any M ∈ k -mod, let C ′ ( M ) = H ( Z /p Z .M ⊗ k p ),and note that the correspondence m m ⊗ p used in (5.6) also provides a k -linear functorial map ψ ∗ : M (1) → C ′ ( M ), where M (1) is the Frobeniustwist. If M = k , then ψ ∗ : k → k is the Frobenius endomorphism. For any M , we can compose ψ ∗ with the Bokstein map and obtain a functorial map(5.10) ϕ ( M ) = ϕ ( M (1) ) → ϕ ( C ′ ( M )[1])of connective spectra. On the other hand, composing (5.9) with the trunca-tion map C q ( Z /p Z , M ⊗ k p ) → C ′ ( M ) also gives a functorial map ψ : ϕ ( M ) → ϕ ( C ′ ( M )[1]), and it suffices to prove that it coincides with (5.10).To do this, note that, unlike the Bokstein map itself, the map (5.10)admits a functorial cone given by the second polynomial Witt vectors functor W of [Ka8]. By definition, this is a functor from k -vector spaces to modulesover the second Witt vectors ring W ( k ) that fits into a functorial short exactsequence(5.11) 0 −−−−→ C ′ ( M ) V −−−−→ W ( M ) R −−−−→ M (1) −−−−→ . If we apply the forgetful functor D ≥ ( W ( k )) → Ho st (Γ), this sequencesinduces an exact triangle of spectra whose connecting differential is (5.10).Equivalently, (5.10) is induced by the composition(5.12) M (1) R − −−−−→ Z ( M ) a −−−−→ C ′ ( M )[1]in D ≥ ( W ( k )), where Z ( M ) is the cone of the map V , the isomorphism R : Z ( M ) → M is induced by R , and a is the natural projection. Moreover, W also fits into a functorial short exact sequence(5.13) 0 −−−−→ M (1) C −−−−→ W ( M ) F −−−−→ C ( M ) −−−−→ , and we have a functorial Teichm¨uller map T : ϕ ( M ) → ϕ ( W ( M )) such that ϕ ( R ) ◦ T = id and ϕ ( F ) ◦ T : M → C ( M ) is the map (5.6).Now let Z = Stab( W ) be the stabilization of the functor W , and notethat by (5.13) and the same argument as in Lemma 5.1, it fits into a func-torial exact triangle(5.14) W ( M ) −−−−→ Z ( M ) a −−−−→ C q ( Z /p Z , M ⊗ k p )[1] −−−−→ D ≥ ( W ( k )) whose connecting differential is the composition of the trun-cation map C q ( Z /p Z , M ⊗ k ) → C ′ ( M ) and the map V of (5.11). In par-ticular, the truncation τ ≤ R ( M ) is exactly Z ( M ) of (5.12), and τ ≤ ( a )is the map a . Moreover, since ϕ ( R ) ◦ T ∼ = id , the stablization Stab( T )of the Teichm¨uller map induces an isomorphism ϕ ( M ) → ϕ ( Z ( M )) in-verse to ϕ ( R ). But on the other hand, we have ψ = ϕ ( F ) ◦ T , so thatStab( ψ ) = ϕ (Stab( F )) ◦ Stab( T ), and the triangles (5.8), (5.14) induce acommutative diagram Z ( M ) −−−−→ Z ( M ) a −−−−→ C ′ ( M )[1] Stab( F ) y τ ≤ Stab( F ) y (cid:13)(cid:13)(cid:13) R ( M ) −−−−→ τ ≤ R ( M ) τ ≤ a −−−−→ C ′ ( M )[1] . Combined with (5.12) and the definiton of ψ , this proves the claim. (cid:3) It turns out that can one connectthe two main examples of addivization of Subsection 5.1 by computing addi-vization of symmetric power functors and their divided power counterparts.If n = p is a prime, this can be done by exactly the same argument as forthe cyclic power functor (5.3) of Lemma 5.1. For other integers, we firstneed an appropriate generalization of Tate cohomology.Fix a commutative ring k . We will say that an admissible family ofsubgroups in a finite group G is a collection X of subgroups H ⊂ G that isclosed under conjugation and intersections, and does not contain G itself.We will also say that a morphism M → N between k [ G ]-modules is X -surjective if M H → N H is surjective for any H ∈ X , and a k [ G ]-module P is X -projective if Hom( P, − ) sends X -surjective maps to surjective maps. Forexample, for any H ∈ X , the induced k [ G ]-module k [ G/H ] = k ⊗ k [ H ] k [ G ]is X -projective by adjunction, and for any k [ G ]-module M projective over k , the product M ⊗ k k [ G/H ] is X -projective by the projection formula. Acomplex M q of k [ G ]-modules is X -exact if M H q is acyclic for any H ∈ X .An X -resolution of a k [ G ]-module M is a complex P q of X -projective k [ G ]-modules, trivial in negative homological degrees and equipped with a map a : P q → M whose cone e P q is X -exact. By exactly the same argumentas in the usual case, a map between two k [ G ]-modules equipped with X -resolutions lifts to a map between resolutions, and the liting is unique up toa chain homotopy. To construct an X -resolution of the trivial k [ G ]-module k , one can take an X -surjective cover η : P → k — for example, by setting P = L H ∈X k [ G/H ] — and then continue by taking P i = P ⊗ k i +10 with the71ifferential as in (4.39). Note that for this particular resolution, the cone e P q is naturally a DG algebra. Definition 5.3.
The truncated Tate cohomology H q ( G, X , M q ) of the group G with respect to the admissible family X and with coefficients in a boundedcomplex M q of k [ G ]-modules is the homology of the complex(5.15) C q ( G, X , M q ) = ( e P q ⊗ k M q ) G , where e P q is the cone of a X -resolution P q → k .By the lifting property of X -resolutions, truncated Tate homology iswell-defined and independent of the choice of the X -resolution P q , and thecomplex (5.15) is well-defined as an object in the derived category D ( k ). Inparticular, we can choose P q in such a way that e P q is a DG algebra, andthis turns H q ( G, X , − ) into a lax monoidal functor. It is also somewhatcohomological, in that a short exact sequence of complexes0 −−−−→ M ′ q −−−−→ M q a −−−−→ M ′′ q −−−−→ X -surjective a gives rise to a long exact sequence of the trun-cated Tate cohomology groups. However, if a is simply surjective, thisneed not be true. In particular, H q ( G, X , − ) does not preserve quasiiso-morphisms, and can be non-trivial even if the complex M q is acyclic. Example 5.4. If X = { e } consists of the trivial subgroup { e } ⊂ G , then X -surjective is surjective, X -exact is exact, X -projective is projective, andfor any k [ G ]-module M , we have H q ( G, { e } , M ) ∼ = τ ≤ ˇ H q ( G, M ) , where as in Lemma 5.1, the right-hand side is the truncation of the usualTate cohomology groups. Example 5.5.
If we have a surjective map of groups f : G → W and an ad-missible family X of subgroups in W , then the family f − X = { f − ( H ) | H ∈X } is an admissible family of subgroups in G . Any k [ W ]-module V definesa k [ G ]-module f ∗ V by restriction of scalars, and for any X -resolution P q of k , f ∗ P q is a f − X -resolution, so that H q ( G, f − X , f ∗ V ) ∼ = H q ( W, X , V ). Remark 5.6.
If one replaces G -invariants in (5.15) with the full cohomol-ogy complexes C q ( G, − ), and takes the sum-total complex of the resulting72icomplex, one arrives at the well-known notion of generalized Tate coho-mology ˇ H q ( G, X , − ) (see e.g. [Ka5, Section 7]). Alternatively, we haveˇ H q ( G, X , M q ) ∼ = RHom q D b ( k [ G ]) / D b X ( k [ G ]) ( k, M q ) , where D b X ( k [ G ]) ⊂ D b ( k [ G ]) is the Karoubi closure of the full triangulatedsubcategory in D b ( k [ G ]) generated by X -projective modules. We alwayshave a map H q ( G, X , M q ) → ˇ H q ( G, X , M q ) but it need not be injective evenwhen M q is k in degree 0. For example, if one takes k = F p and G = Z /p Z ,with X consisting of Z /p Z ⊂ Z /p Z , then H q ( G, X , k ) ∼ = H q ( Z /p Z , k ) byExample 5.5, while ˇ H q ( G, X , k ) = 0.To study truncated Tate cohomology, it is convenient to do the following.Let Γ G be the category of finite G -sets — that is, finite sets S equipped withan action of G — and let O G ⊂ Γ G be the full subcategory spanned by G -orbits , that is, G -sets S such that the action is transitive. For any subgroup H ⊂ G , the quotient G/H is a G -orbit that we denote by [ G/H ] ∈ O G ⊂ Γ G ,and all G -orbits are of this form. The category Γ G has finite products andfinite coproducts. Say that a functor E ∈ Fun(Γ G , k ) is additive if for any S, S ′ ∈ Γ S , the natural map E ( S ) ⊕ E ( S ′ ) → E ( S ⊔ S ′ ) is an isomorphism,and let Fun add (Γ G , k ) ⊂ Fun(Γ G , k ) be the full subcategory spanned byadditive functors. Then we have a natural equivalence(5.16) Fun( O G , k ) ∼ = Fun add (Γ G , k )given by the pullback ε ∗ with respect to the embedding ε : O G → Γ G , withthe inverse equivalence given by the left Kan extension ε ! .For any subgroup H ⊂ G , the left comma-fiber Γ G / [ G/H ] is naturallyidentified with Γ H , and the forgetful functor ψ H : Γ H ∼ = Γ G / [ G/H ] → Γ G has a right-adjoint µ H : Γ G → Γ H sending S ∈ Γ G to S × [ G/H ] with itsnatural projection to [
G/H ]. Then by adjunction, we have(5.17) µ ∗ H ∼ = ψ H ! , so that ψ H ! is exact. The functor µ H commutes with finite coproducts,so that (5.17) sends Fun add (Γ H , k ) ⊂ Fun(Γ H , k ) into Fun add (Γ G , k ) ⊂ Fun(Γ G , k ). The functor ψ H sends orbits to orbits, thus restricts to a functor ψ H : O H → O G , and this is compatible with the equivalences ε ! of (5.16),so that the left Kan extension ψ H ! : Fun( O H , k ) → Fun( O G , k ) is also exact.If G is equipped with an admissible family of subgroups X , then we canconsider the full subcategory O X ⊂ O G spanned by orbits [ G/H ] with H X . Since by assumption, X does not contain G itself, we can extendthe embedding O X → O G to a full embedding O > X → O G by sending theterminal object o ∈ O > X to the one-point G -orbit [ G/G ], and we let Γ X ⊂ Γ G be the full subcategory spanned by finite coproducts of orbits in O > X ⊂ O G .Then Γ X ⊂ Γ G is closed under finite products. We still have the notion ofan additive functor in Fun(Γ X , k ) and the equivalence (5.16), and for any H in X , we still have the functor ψ H : O H → O X ⊂ O > X such that the left Kanextension ψ H ! : Fun( O H , k ) → Fun( O > X , k ) is exact and given by (5.17).Now, let σ : Γ oG → k [ G ]-mod be the tautological functor sending a G -set S to the space k ( S ) of k -valued functions on S . Restricting σ to Γ o X ⊂ Γ oG and taking its left Kan extension with respect to the Yoneda embedding(1.18) gives a right-exact functor Y ! ( σ ) : Fun(Γ X , k ) → k [ G ]-mod that hasa right-adjoint Loc X : k [ G ]-mod → Fun(Γ X , k ). Explicitly, Loc X is given by(5.18) Loc X ( M )( S ) = ( M ⊗ k k [ S ]) G , S ∈ Γ X , M ∈ k [ G ]-mod , so that in particular, it takes values in the subcategory Fun add (Γ X , k ). Weextend Loc X to bounded complexes by applying it pointwise, so that for anybounded complex M q ∈ C b q ( k [ G ]), we have a well-defined object Loc X ( M q )in the bounded derived category D b ( O > X , k ). Lemma 5.7.
For any bounded complex M q of k [ G ] -modules, we have H q ( G, X , M q ) ∼ = H q ( O > X , { o } , Loc X ( M q )) where H q ( − ) is the homology with support of (1.30) .Proof. Let σ ∗ : Γ G → k [ G ]-mod be the functor sending S ∈ Γ G to the free k -module k [ S ] (that is, to the dual k -module to k ( S )), with the left Kanextension L = Y ! ( σ ∗ ) : Fun( O >o X , k ) → k [ G ]-mod, and note that for any[ G/H ] ∈ O
Loc H is the functor (5.18) for the category Γ H , and by (1.26), it thensuffices to observe that ψ Ho ∗ k { o } = 0. (cid:3) Recall that we denote by Γ − the category of non-empty finite sets and surjective maps. For any integer n ≥
1, let [ n ] ∈ Γ − be the set of cardinality n , and let Σ n = Aut([ n ]) be the correspondingpermutation group. For any flat k -module V , the n -th symmetric power S n ( V ) and divided power D n ( V ) are given by(5.21) S n ( V ) = T n ( V ) Σ n , D n ( V ) = T n ( V ) Σ n , where T n ( V ) = V ⊗ k n is the tensor power functor. For any n, m ≥
1, theisomorphism T m ( T n ( V )) ∼ = T mn ( V ) induces functorial maps(5.22) S m ( S n ( V )) → S mn ( V ) , D mn ( V ) → D m ( D n ( V )) . For any map f : [ n ] → [ m ] in Γ − , let Σ f ⊂ Σ n be the subgroup of automor-phisms a : [ n ] → [ n ] such that f ◦ a = f . Explicitly, f defines a partition[ n ] = [ n ] ⊔ . . . ⊔ [ n m ] of [ n ] into [ m ] disjoint subsets [ n i ] = f − ( i ), i ∈ [ m ],and Σ f = Σ [ n ] × · · · × Σ [ n m ] consists of permutations that preserve each [ n i ].For any [ n ], let X n be the family of subgroups Σ f ⊂ Σ n for all f : [ n ] → [ m ]in Γ n with m ≥
2. Then the family X n is admissible. The augmented or-bit category O > X n is naturally identified with the category Γ n whose objectsare maps f : [ n ] → [ m ] in Γ − , and whose maps from f ′ : [ n ] → [ m ′ ] to f : [ n ] → [ m ] are maps g : [ m ′ ] → [ m ] such that g ◦ f ′ factors through f (butwe do not specify a factorization, so that Γ n ( f ′ , f ) ∼ = ar (Γ − )( f ′ , f ) / Σ f ). Remark 5.8.
The standard notation for the divided power functor is Γ n and not D n ; we allow ourselves to change it to avoid confusion with thenotation for the category of finite sets.For any n , the divided power functor D n of (5.21) is a functor from k -mod fl to k -mod, and it is lax monoidal by adjunction, thus defines a laxmonoidal functor(5.23) Q n q = Stab( D n ) : k -mod fl → C ≥ ( k ) , V ∈ k -mod fl , we denote by HQ n q ( V ) the homology of the complex Q n q ( V ). If we take V = k , then Q n q ( k )is a DG algebra that defines a commutative ring object in Ho( k ) ∼ = D ≥ ( k ),and HQ q ( k ) is a graded-commutative algebra over k . Lemma 5.9.
For any n ≥ , we have a natural isomorphism (5.24) Q n q ( V ) ∼ = C q (Σ n , X n , T n ( V )) of commutative ring objects in D ≥ ( k ) whose target is as in (5.15) .Proof. Choose an X n -resolution P q of the trivial k [Σ n ]-module k such thateach P i is a sum of modules k [Σ n / Σ f ], Σ f ∈ X n , and then proceed exactlyas in Lemma 5.1. On one hand, for each f : [ n ] → [ m ], we have(5.25) ( T n ( V ) ⊗ k k [Σ n / Σ f ]) Σ n ∼ = T n ( V ) Σ f ∼ = O ≤ i ≤ m D n i ( V ) , and since m ≥
2, this has trivial stabilization by (4.15). On the other hand,the quotient T n ( V ⊕ V ) / ( T n ( V ) ⊕ T n ( V )) is X n -projective, so that theright-hand side of (5.24) is already stable. (cid:3) With a little bit of extra effort, one can show that (5.24) lifts to a quasi-isomorphism between DG algebras, but we will not need this: our maininterest lies in the homology algebra HQ n q ( k ). To compute it, recall thatLemma 5.7 provides an isomorphism(5.26) H q (Σ n , X n , T n ( V )) ∼ = H q (Γ n , { o } , L n ( V )) , where { o } ∈ Γ n is the terminal object [ n ] → [1], and we simplify notationby writing L n = Loc X n ◦ T n . Note that for any p ≥
1, the cartesian productof finite sets induces a product functorΓ n × Γ p → Γ np , and in particular, taking the product with the terminal object [ p ] → [1] givesa functor ε p : Γ n → Γ pn . This is a fully faithful embedding whose essentialimage consists of partitions [ pn ] = [ d ] ⊔ . . . ⊔ [ d l ] such that every d i , 1 ≤ i ≤ l is divisible by p . It is also right-closed in the sense of Example 1.15. For anybounded complex M q of flat k -modules, (5.18), (5.22) and (5.25) provide anatural map(5.27) ε ∗ p L np ( M q ) → L n ( D p ( M q ))76hat is an isomorphism if M = k or M = k [1].Now let I q be the acyclic length-2 complex id : k → k placed in homo-logical degrees 0 and 1, so that fits into a short exact sequence(5.28) 0 −−−−→ k b −−−−→ I q a −−−−→ k [1] −−−−→ , and let I ∗ q = I q [ −
1] be the same complex id : k → k placed in degrees 0 and −
1. Moreover, assume from now on that k is annihilated by a prime p . Lemma 5.10.
The complex D m ( I q ) is acyclic unless m = np is divisibleby p , and in the latter case, the map ϕ n : D np ( I q ) → D n ( D p ( I q )) inducedby (5.22) is a quasiisomorphism. Moreover, if p is odd, then this map is anisomorphism, and D m ( I ∗ q ) is acyclic for any m .Proof. By duality, we may replace D m with S m , replace ϕ n with the dualmap ϕ ∗ n , and swap I q and I ∗ q . If p is odd, then the symmetric algebra S q ( I q )resp. S q ( I ∗ q ) is the free graded-commutative algebra k [ t, ξ ] with deg t = 0,deg ξ = 1 resp. −
1, and the differential dξ = t resp. dt = ξ . The claimis then obvious. If p = 2, then we still have S q ( I ∗ q ) ∼ = k [ t, ξ ], deg ξ = − dt = ξ , but “graded-commutative” now means commutative, so that ϕ ∗ n isnot an isomorphism anymore. However, it is trivial to check that k [ t, ξ ] isacyclic in degrees other than 0, and its homology in degree 0 is k [ t ]. (cid:3) Remark 5.11. If m is not divisible by p , then Lemma 5.10 provides a map k → D m ( I q ) ⊂ T m ( I q ) that splits the differential D m ( I q ) → D m ( I q ) ∼ = D m ( k ) ∼ = k , so that T m ( I q ) ∼ = k ⊕ M for some Σ m -module M flat over k . Then T m ( I q ) ∼ = I q ⊗ k S q ( M [1]) is Σ m -equivariantly contractible, so that D m ( V ⊗ I q ) is acyclic for any flat k -module V .Since the embedding ε p : Γ n → Γ pn is right-closed for any n ≥
1, the leftKan extension ε p ! is given by extension by 0. Then by (5.18) and (5.25),Lemma 5.10 immediately implies that the adjunction map ε p ! ε ∗ p L np ( I q ) → L np ( I q ) is a quasiisomorphism. Moreover, if p is odd, then (5.27) is an iso-morphism for M q = I q . The complex I q = D p ( I q ) is isomorphic to k ⊕ k [1],and the isomorphism inverse to (5.27) induces by adjunction a quasiisomor-phism(5.29) ε p ! L n ( I q ) → L np ( I q )that gives rise to an isomorphism(5.30) HQ n q ( I q )) → HQ np q ( I q )77n D ≥ ( k ). If p = 2, then D p ( I q ) is quasiisomorphic to k placed in degree 0,so we may let I q = k , construct a quasiisomorphism (5.29) by composing theadjunction map ε p ! L n ( k ) → L pn ( k ) with the map induced by the embedding b : k → I q of (5.28), and still obtain an isomorphism (5.30). Moreover, if weturn I q and I q into DG algebras with trivial multiplication on I and I , sothat b in (5.28) is a DG algebra map, then (5.30) is an algebra isomorphism. Lemma 5.12.
For any commutative ring k annhilated by an odd prime p ,and any n ≥ , we have an isomorphism of graded-commutative algebras (5.31) HQ np q ( k ) ∼ = HQ n q ( k )[ ξ, τ ] , where the generators ξ , τ have degrees deg ξ = 2( pn − , deg τ = 2 n − . If p = 2 , we have the same isomorphism but without τ and with deg ξ = 2 n − .Proof. For any m ≥
1, the sequence (5.28) induces a sequence(5.32) 0 −−−−→ T m ( k ) b −−−−→ T m ( I q ) a −−−−→ T m ( k [1]) −−−−→ k [Σ m ]-modules. This sequence is termwise-split, and whileit is not exact in the middle term, the homology there is X m -projective ineach degree. Therefore (5.32) gives rise to an exact triangle(5.33) Q m q ( k ) β −−−−→ Q m q ( I q ) α −−−−→ Q m q ( k [1]) δ −−−−→ in D ≥ ( k ). All maps here are maps of Q m q ( k )-modules, and β is also analgebra map. If p = 2, we have T m ( k [1]) ∼ = T m ( k )[ m ], so that Q m q ( k [1]) ∼ = Q m ( k )[ m ]. If p is odd, we can shift the sequence (5.28) and repeat theargument to obtain an exact triangle(5.34) Q m q ( k [ − −−−−→ Q m q ( I ∗ q ) −−−−→ Q m q ( k ) −−−−→ in D ≥ ( k ), where L m ( I ∗ q ), hence also Q m q ( I ∗ q ) vanishes by (5.18), (5.25) andLemma 5.10. Therefore the connecting differential in the triangle (5.34) isan isomorphism, and since T m ( k [1]) ∼ = T m ( k [ − m ], we have Q m q ( k [1]) ∼ = Q m q ( k [ − m ] ∼ = Q m q ( k )[2 m − m = pn , then we have theidentification (5.30) induced by the quasiisomorphism (5.29), and in termsof this identification, the map α in (5.33) is induced by a map ε p ! L n ( I q ) → L pn ( k [1]) adjoint to a map α ′ : L n ( I q ) → ε ∗ p L pn ( k [1]) ∼ = L n ( D p ( k [1])). If p is odd, then D p ( k [1]) = 0, and if p = 2 and D p ( k [1]) ∼ = k [ p ], we have α ′ = L n ( D p ( a ◦ b )) = 0 since a ◦ b = 0. Thus in any case, α ′ vanishes,78o does α , and the triangle (5.33) splits after one rotation. Altogether, weobtain a short exact sequence0 −→ HQ np q − d ( k ) δ −→ HQ np q ( k ) β −→ HQ n q ( I q ) −→ HQ np q ( k )-modules, where d = 2 m − np −
1) if p is odd and d = np − n − p = 2. Moreover, β is an algebra map. Since δ isa module map, it must be given by multiplication by the element ξ = δ (1)of degree d , and if p = 2, I q = k and we are done. If p is odd, then tofinish the proof, it remains to construct a commutative algebra isomorphism HQ n q ( I q ) ∼ = HQ n q ( k )[ τ ], where τ is a generator of degree 2 n −
1. To dothis, note that I q also fits into an exact sequence of the form (5.28) thatis moreover split, and then the triangle (5.32) for m = n induced by thissequence is also split. (cid:3) Now assume that k = F p is a primefield. Then for any k -vector space V , we have a natural k -linear map V → S p ( V ) sending v ∈ V to v ⊗ p , and for any n ≥
1, this can be combined with(5.22) to give a functorial map(5.35) S n ( V ) −−−−→ S p ( S n ( V )) −−−−→ S pn ( V ) . If V is finite-dimensional, we have the dual map(5.36) D pn ( V ) −−−−→ D p ( D n ( V )) −−−−→ D n ( V ) , and we extend it to all vector spaces by taking filtered colimits. Beingfunctorial, the map (5.36) induces maps(5.37) Q pn q ( V ) → Q n q ( V ) , HQ pn q ( V ) → HQ n q ( V )of the stabilizations (5.23) of functors (5.21) and their homology modules.Note that the map (5.36) is compatible with the lax monoidal structures, sothat (5.37) is also a lax monidal map. Lemma 5.13.
In terms of the isomorphism (5.31) , the map (5.37) is ob-tained by sending the generators τ and ξ to .Proof. Let I q and b be as in (5.28), and note that for any k -vector space V ,the map (5.36) for n = 1 factors as(5.38) D p ( V ) D p ( b ) −−−−→ D p ( V ⊗ I q ) d −−−−→ V d . Therefore we have a functorial diagram(5.39) L pn ( V ) L pn ( b ) −−−−→ L pn ( V ⊗ I q ) e ←−−−− ε p ! ε ∗ p L pn ( V ⊗ I q ) y ε p ! L n ( V ) L n ( d ) ←−−−− ε p ! L n ( D p ( V ⊗ I q ))of complexes in Fun(Γ pn , k ), where the vertical map is (5.27), and e is theadjunction map. By Remark 5.11, e is a quasiisomorphism, thus invertible in D ≥ (Γ pn , k ), so that (5.39) defines a map L pn ( V ) → ε p ! ( V ) in D ≥ (Γ pn , k ).After evaluation at { o } ∈ Γ pn , this map becomes (5.36). Being functo-rial, the diagram (5.39) also defines a diagram in Ho( k -mod × Γ pn , k ) ∼ = D ≥ ( k -mod × Γ pn , k ), where e is again invertible. Then if we denote by z : k -mod → k -mod × Γ pn the right-closed embedding onto k -mod × { o } ,and apply the functor L q z ! of (1.19) together with the identifications (5.26)and (5.24), we obtain a map Q pn → Q n in Ho( k -mod , k ) that fits into acommutative diagram D pn −−−−→ D n y y Q pn −−−−→ Q n , where the top arrow is (5.36), and the vertical arrows are stabilization maps.By the universal property of stabilization, the bottom arrow then mustcoincide with (5.37). To finish the proof, it remains to evaluate at V = k ,and compare (5.39) with the construction of the isomorphism (5.31) givenin Corollary 5.12. (cid:3) Now, Lemma 5.12 immediately implies by induction that Q n q ( k ) = 0unless n is a power of p , and then being a Q n q ( k )-module, Q n q ( V ) must vanishfor any k -vector space V . Thus we might as well renumber the functors Q n q by setting(5.40) Q ( i ) q = Q p i q , HQ ( i ) q = HQ p i q , i ≥ , and let D ( i ) = D p i . Since k is a prime field, the commutative algebra k ( V ) of all k -valued functions on a k -vector space V has the Frobeniusendomorphism equal to the identity, f p = f , so that the tautological map V ∗ → k [ V ] extends to maps S n ( V ∗ ) → k ( V ), n ≥ k [ V ] → D n ( V ) that give riseto a map(5.41) k [ V ] → lim i D ( i ) ( V ) , V , so that it induces a map(5.42) Q q ( k/k ) → R q lim i Q ( i ) q ( k ) , where as in Subsection 5.1, Q q ( k/k ) is the stabilization of the linear spanfunctor V k [ V ]. Lemma 5.14.
The map (5.42) is a quasiisomorphism.Proof.
To construct stabilizations in Ho(
I, k ) for some half-additive cat-egory I , we only need to consider finite coproducts in I . Therefore wecan restrict our attention to the subcategory P ( k ) ⊂ k -mod spanned byfinite-dimensional k -vector spaces V . For any such V , denote by K q ( V ) = S q ( V ⊗ k I q ) the total symmetric power of the complex V ⊗ k I q , where I q is as (5.28). Then K q ( V ) is a flat S q ( V )-algebra quasiisomorphic to k (this is the standard Koszul resolution). Moreover, any k -linear map a : V → S q ( V ) uniquely extends to an algebra map exp( a ) : S q ( V ) → S q ( V ), and we can consider the complex K q ( V, a ) = K q ( V ) ⊗ S q ( V ) S q ( V ),where S q ( V ) is a module over itself via the map exp( a ). We have a mapexp( a ) = id ⊗ exp( a ) : K q ( V ) → K q ( V, a ), and if exp( a ) is flat, the complex K q ( V, a ) only has homology in degree 0. In particular, this is the case if a = ϕ : V → S p ( V ) ⊂ S q ( V ) is the map (5.35), and if a = ϕ − id is its dif-ference with the identity map id : V → V ⊂ S q ( V ). Moreover, in the lattercase, we can equip K q ( V, ϕ − id ) with a multiplicative increasing filtration F q by assigning filtered degree 1 to V ⊂ S q ( V ) = K ( V, ϕ − id ) and p to V ⊗ k ⊂ V ⊗ S q ( V ) = K ( V, ϕ − id ). Then gr F q K q ( V, ϕ − id ) ∼ = K q ( V, ϕ ), sothat all the associated graded pieces also have homology only in degree 0,and then by the spectral sequence argument, the same holds for the filteredpieces F n K ( V, ϕ − id ), n ≥ V is finite-dimensional and k is prime, all k -valued functions onthe dual vector space V ∗ are polynomial, so that the map S q ( V ) → k ( V ∗ ) issurjective. In effect, k ( V ∗ ) is the quotient of S q ( V ) by the ideal generatedby ( ϕ − id )( V ), and we have a quasiisomorphism K q ( V, ϕ − id ) ∼ = k ( V ∗ ).Dually, for any map a : V ∗ → S q ( V ∗ ), let K q ( V, a ) = K q ( V ∗ , a ) ∗ , with thedescreaing filtration F n K q ( V, a ) = F n K q ( V ∗ , a ) ∗ ; then we have a functorialquasiisomorphism(5.43) k [ V ] ∼ = K q ( V, ϕ − id ) ∼ = lim n F n K q ( V, ϕ − id ) , where all the terms in the limit have homology concentrated in degree 0.The induced filtration on the group algebra k [ V ] is the filtration by the81owers of the augmentation ideal, and since V is finite-dimensional, it goesto 0 at some finite step. Moreover, if we assign filtered degree p i to D ( i ) ( V ),then (5.41) becomes a filtered map, and by Remark 4.28, additivization withrespect to V commutes with the limits in (5.41) and in (5.43). Thereforeit suffices to prove that the associated graded quotient of the map (5.41)becomes a quasiisomorphism after applying additivization.To do this, it is convenient to lift (5.41) to a map of complexes. On onehand, we have the surjective map exp( ϕ − id ) ∗ : K q ( V, ϕ − id ) → K q ( V ), itbecomes filtered if the rescale the filtration on its target by p , and K q ( V ) ∼ = k has trivial additivization on all its graded pieces, so that we may replace K q ( V, ϕ − id ) in (5.43) with K q ♭ ( V ) = Ker exp( ϕ − id ) ∗ . On the other hand,by the telescope construction, we have an exact sequence(5.44) 0 −→ lim i D ( i ) V −→ Y i ≥ D ( i ) ( V ) ϕ ∗ − id −→ Y i ≥ D ( i ) ( V ) −→ , where ϕ ∗ : D ( i +1) ( V ) → D ( i ) ( V ) are the maps (5.36), and (5.44) becomes afiltered exact sequence if we assign filtered degree p i resp. p i +1 to D ( i ) ( V ) inthe middle resp. rightmost term. But for any a , we have a natural projection K q ( V, a ) → K ( V, a ) ∼ = Y n D n ( V ) → Y i D ( i ) ( V ) , and it intertwines exp( ϕ − id ) ∗ and the map ϕ ∗ − id in (5.44), thus inducesa filtered map K q ♭ ( V ) → lim i D ( i ) ( V ). This is our map.It now remains to observe that gr nF K i ( V, ϕ − id ) = gr nF K i ( V ) = 0 if ip > n , and exp( ϕ − id ) ∗ : gr nF K i ( V, ϕ − id ) → gr nF K i ( V ) is an isomorphismif n = ip , so that gr nF K i♭ ( V ) = 0 when ip ≥ n . If 0 < ip < n , then gr nF K i ( V, ϕ − id ) ∼ = Λ i ( V ) ⊗ D n − ip ( V ) has trivial additivization by (4.15),and similarly for gr nF K i ( V ), and if i = 0 but n is not a power of p , thenboth have trivial additivization by Lemma 5.12. Finally, if n = p i , then gr nF K ♭ ( V ) is the n -th associated graded piece of (5.44) on the nose. (cid:3) To see how (5.42) yields (5.1), note that by Lemma 5.12 and induction,we have(5.45) HQ ( i ) q ( k ) ∼ = k [ β, ξ , τ , . . . , ξ i − , τ i − , ξ i ] , where the degrees of the generators are the same as in (5.1), there are no τ if p = 2, and the transition maps in (5.42) act on homology by sendingthe extra generators to 0. Therefore the inverse system of homology groupsstabilizes at a finite step in each degree, R lim vanishes, and the homology HQ q ( k/k ) of the DG algebra Q q ( k/k ) is exactly as in (5.1).82 emark 5.15. The dual Steenrod algebra is of course not just an algebrabut a Hopf algebra, with some comultiplication that is too non-linear to ad-mit a DG model. In terms of stabilization, one observes that by adjunction,the linear span functor V k [ V ] is a comonad, and then the endofunc-tor of the category Ho( k ) given by its stabilization Q q is also a comonad.Since the comonad is non-linear, adding an enhancement to it requires sometechnology, but whatever technology one uses, enhanced coalgebras overthis enhanced comonad are connective spectra, for more-or-less tautologicalreasons (for example, if one uses “stable model pairs”, then this is [Ka9,Theorem 10.6]). However, observe that the whole projective system (5.41)has a structure of a comonad, with the structure maps (5.22), and then sodoes its stabilization (5.42). In other words, the filtration on Q q given by(5.42) is compatible with the comonad structure, and then coalgebras overthis filtered version of Q q form a “filtered” version of the stable homotopycategory. It seems that this category has not been considered yet, and itmight be interesting. We hope to return to this elsewhere. -categories. -categories. Let us now recall the de-scription of symmetric monoidal structures in terms of the category Γ + givenin Subsection 4.4. To encode non-symmetric monoidal categories, one canuse the same Segal machine as in Definition 4.10 but with ∆ o instead of Γ + .It is convenient to start with a more general notion of a 2-category. Definition 6.1.
A cofibration
C → ∆ o satisfies the Segal condition if forany n ≥ l ≥
0, the functor(6.1) C [ n ] → C [ l ] × C [0] C [ n − l ] induced by (3.6) is an equivalence. A 2 -category is a cofibration C → ∆ o with discrete C [0] satisfying the Segal condition. A lax -functor between2-categories C , C ′ is a functor γ : C → C ′ over ∆ o cocartesian over anchormaps. A 2 -functor is a lax 2-functor that is cocartesian over all maps. Remark 6.2.
More generally, for any n ≥ l ′ ≥ l ≥
0, we have a cocartesiansquare(6.2) [ l ′ − l ] t −−−−→ [ l ′ ] s y y s [ n − l ] t −−−−→ [ n ]83n ∆, and if C → ∆ o satisfies the Segal condition, then we also have C [ n ] ∼ = C [ l ′ ] × C [ l ′− l ] C [ n − l ] for any square (6.2) (just combine (6.1) for n ≥ l ′ ≥ l ′ ≥ l ≥ Remark 6.3.
Assume given a cofibration π : C → ∆ o , and say that afunctor c : V → C o is standard if it is cartesian over ∆, and π o ◦ c : V → ∆ o is the top left part of a square (6.2). Then C satisfies the Segal condition ofDefinition 6.1 if and only if for any standard c : V → C o there exists a colimitcolim V c o , and the natural map π (colim V c o ) → colim V ( π o ◦ c ) = [ n ] is anisomorphism. Moreover, since s : [ l ] → [ n ] and t : [ n − l ] → [ n ] are anchormaps, the opposite γ o to any lax 2-functor γ : C → C ′ to some 2-category C ′ preserves the colimits of standard functors c : V → C o . In addition tothis, say that c : V → C o is half-standard if c ( o ) → c (0) is a cartesian liftingof an anchor map, and c ( o ) → c (1) is vertical with respect to π o . Then forsuch a c , colim V c exists as soon as C satisfies the Segal condition, and thesecolimits are also preserved by opposites γ o to lax 2-functors. Example 6.4.
For any category C , let ε : pt → ∆ o be the embedding onto[0], and let E C = ε ∗ C . Then the fibers of the cofibration E C → ∆ o are E C [ n ] ∼ = C V ([ n ]) , so that E C trivially satisfies the Segal condition, and if C isdiscrete, then E C is a 2-category. Moreover, for any cofibration π : C → ∆ o ,we have a natural functor(6.3) ν : C → E C [0] = ε ∗ ε ∗ C induced by (2.22). This functor ν is cocartesian over ∆ o , and if C is a2-category, ν is a 2-functor. Remark 6.5.
The second condition of Definition 6.1 – namely, the require-ment that C [0] is discrete – can be always achieved by the following trick.For any cofibration C → ∆ o , consider the discrete subcategory C [0] , Id ⊂ C [0] ,and define the reduction C red by the cartesian square C red −−−−→ C y y ν E C [0] , Id −−−−→ E C [0] , where ν is the functor (6.3). Then if C satisfies the Segal condition, so does C red , and ( C red ) [0] = C [0] , Id is discrete. However, this procedure has to be84sed with caution, since C [0] , Id ⊂ C [0] , hence also C red depend on C on thenose, and not only on its equivalence class. To aleviate the problem, it isbetter to first replace C with its tightening.Definition 6.1 is a convenient packaging of the usual notion of a (weak)2-category: objects are objects of C [0] , and for any c, c ′ ∈ C [0] , the fiber C ( c, c ′ ) of the projection s o ! × t o ! : C [1] → C [0] over c × c ′ is the category ofmorphisms from c to c ′ . For m ≥
2, we have an equivalence(6.4) m Y i =1 a oi ! : C [ m ] ∼ = C [1] × C [0] · · · × C [0] C [1] , with m copies of C [1] numbered by edges i ∈ E ([ n ] δ ) = { , . . . , m } of thestring quiver [ n ] δ , and a i : [1] → [ m ] standing for the embeddings of theedges. For every c ∈ C [0] , we have the identity object id c ∈ C ( c, c ) inducedby the tautological projection [1] → [0], and the composition functors − o − : C ( c, c ′ ) × C ( c ′ , c ′′ ) → C ( c, c ′′ ) are induced by the map m : [1] → [2] sending 0to 0 and 1 to 2.The point -category pt is ∆ o itself. As it should, it has one object, pt ∼ = pt , and for any object c ∈ C [0] in a 2-category C , the embedding ε ( c ) : pt → C [0] onto c uniquely extends to a 2-functor ε ( c ) : pt → C thatwe also call the embedding onto c . The tautological projection τ : C → pt is the structural cofibration. If we have another 2-category C ′ , then the constant -functor C ′ → C with value c is the composition of τ : C ′ → pt and ε ( c ) : pt → C . The 2 -product C × C ′ is the product C × ∆ o C ′ . A2-category C is discrete if C ∼ = C [0] × ∆. For any 2-category C , applying theinvolution ι : ∆ → ∆, [ n ] [ n ] o gives a cofibration ι ∗ C → ∆, and this isalso a 2-category; we call it the opposite -category and denote C ι .In keeping with our usage for ordinary categories, we say that a lax 2-functor γ : C ′ → C is dense if γ [0] : C ′ [0] → C [0] is an equivalence. For any lax2-functor γ , we can define a 2-category γ ∗ C by the cartesian square(6.5) γ ∗ C −−−−→ C y y ν E C ′ [0] −−−−→ E C [0] , where ν is as in Example 6.4. Then γ factors as(6.6) C ′ e γ −−−−→ γ ∗ C γ −−−−→ C , where e γ is tautologically dense. We say that γ is 2 -fully faithful if e γ is anequivalence; in particular, γ in (6.6) is tautologically 2-fully faithful.85 xample 6.6. If γ is the tautological embedding γ : C [0] × ∆ o → C , then γ ∗ C ∼ = C . More generally, if we have an embedding S ⊂ C [0] , with thecorresponding functor γ : S × ∆ o → C , then γ ∗ C can be thought of a the full2-subcategory in C spanned by objects c ∈ S ⊂ C [0] .It is useful to generalize Example 6.6 in the following way. Assumegiven a functor S : C [0] → Sets (that is, a set S c for any c ∈ C [0] ). Denote by π : C [0] [ S ] → C [0] be the corresponding discrete cofibration, let χ : C [0] [ S ] × ∆ o → C be the composition of the 2-functor π × id : C [0] [ S ] × ∆ → C [0] × ∆with the tautological embedding C [0] × ∆ o → C , and let(6.7) C [ S ] = χ ∗ C , where the right-hand side is as in (6.5). Equivalently, C [ S ] can by obtainedby extending S to a functor S : C →
Sets by the right Kan extension withrespect to the embedding C [0] → C , and taking the corresponding discretecofibration C [ S ] → C . Explicitly, for any [ n ] ∈ ∆ and c ∈ C [ n ] , we have S ( c ) ∼ = n Y i =0 S ( b oi ! c ) , where b i : [0] → [ n ] sends 0 to i . One can also consider the universalsituation: the forgetful functor Sets + → Sets is a discrete cofibration withfiber S over any S ∈ Sets, we can consider the induced cofibration E Sets + → E Sets, and then C [ S ] fits into a cartesian square(6.8) C [ S ] −−−−→ E Sets + π y y C E ( S ) ◦ ν −−−−→ E Sets , where ν is the functor (6.3). Note that up to an isomorphism, the functor S factors through Sets Id , so that we may replace E Sets + → E Sets in (6.8)with the induced cofibration over E Sets Id ⊂ E Sets without changing C [ S ](since the cofibration E Sets Id → ∆ o is discrete, E ( S ) ◦ ν in (6.8) then alsobecomes a cofibration). In either description, we obviously have C [ pt ] ∼ = C ,where pt : C [0] → Sets sends everything to the one-element set.
In any meaningful for-malism, a usual category should define a 2-category. In the context of Defi-nition 6.1, this can be achieved as follows. For any category I , denote(6.9) ∆ o h I i = Id + ∗∗ ( I × ∆ o ) , I × ∆ o → ∆ o is the trivial cofibration, Id : ∆ o → ∆ o is the identityfunctor, + is the class of special maps, and Id + ∗∗ has the same meaning asin Example 2.14. If we identify ar t (∆) and the category ∆ q of (3.7), then(2.28) and (2.31) provide an identification ∆ o h I i ∼ = Fun(∆ q / ∆ , I ), so thatfor any [ n ] ∈ ∆ o , the fiber ∆ o h I i [ n ] of the cofibration ∆( I ) → ∆ o is thefunctor category Fun([ n ] , I ). In particular, ∆ o h I i obviously satisfies theSegal condition. Definition 6.7.
The simplicial replacement ∆ o I of a category I is the re-duction ∆( I ) red of the cofibration (6.9) in the sense of Remark 6.5.Explicitly, objects in ∆ o I are pairs h [ n ] , i q i of an object [ n ] ∈ ∆ and afunctor i q : [ n ] → I , with maps from h [ n ] , i q i to h [ n ′ ] , i ′ q i given by a map f : [ n ′ ] → [ n ] and a map i ′ q → f ∗ i q that is pointwise an identity map.The augmented simplicial replacement is the category ∆ o> I = (∆ o I ) > . Ifthe category I is small, ∆ o I ∼ = ∆ o N ( I ) and ∆ o> I = ∆ o> N ( I ) are thecategories of simplicis of its nerve N I : ∆ o → Sets, as in Subsection 3.4. Forany I , ∆ o I is a 2-category in the sense of Definition 6.1, with the structuralcofibration ∆ o I → ∆ o given by the forgetul functor h [ n ] , i q i 7→ [ n ], and thecofibration is discrete. Its extension ∆ o> I → ∆ o> = ∆ The correspondence I ∆ o I respects objects and mor-phisms but loses the 2-categorical structure. In particular, for an equiv-alence I ′ → I between small categories, the induced functor ∆ o I ′ → ∆ o I is87ot in general an equivalence. In terms of (6.7), we have ∆ o I ′ ∼ = ∆ o I [ S ],where S : I Id → Sets sends i ∈ I to the set of its preimages in I ′ .Another useful class of 2-categories is (1 , -categories , namely, 2-cate-gories C such that for any c, c ′ ∈ C [0] , the category C ( c, c ′ ) is a groupoid.Equivalently, one can require that the structural cofibration C → ∆ o issemidiscrete. For any 2-category C , the dense subcategory C ♮ ⊂ C spannedby cocartesian maps is then a (1 , C ( c, c ′ ) Iso ⊂ C ( c, c ′ ) as categories of morphisms. If C ♮ is bounded, we can consider its truncation τ ( C ♮ ). Objects in τ ( C ♮ ) are stillobjects c ∈ C [0] , and maps are isomorphism classes of 1-morphisms in C .By abuse of terminology, we define a 2-functor from a category I to a2-category C as a 2-functor γ from ∆ o I to C , and similarly for lax 2-functors.In the other directions, since ∆ o I is discrete, any lax 2-functor from C to∆ o I is automatically a 2-functor. It turns out that these then correspondbijectively to usual functors C → I of a special kind. Definition 6.10. A morphism f in a 2-category C is special if it is a cocarte-sian lifting of a special map in ∆ o . A functor E : C → E to some category E is special if it inverts all special maps, and a cofibration C ′ → C is special if for any special map f in C , the transition functor f ! is an equivalence.For any small category I with simplicial replacement ∆ o I , the evaluationfunctor (2.23) induces a special functor(6.10) ξ : ∆ o I → I that sends h [ n ] , i q i to i q (0) ∈ I . We also have ∆ o I o ∼ = ι ∗ ∆ o I , so that (6.10)for the category I o provides a functor ξ ⊥ : ∆ o I → I o sending h [ n ] , i q i to i q ( n ) ∈ I o . One can also combine ξ and ξ ⊥ into a single functor(6.11) ξ ♭ : ∆ o I → tw ( I )sending h [ n ] , i q i to the arrow i q (0) → i q ( n ). Then ξ and ξ ⊥ are obtained bycomposing (6.11) with the projections (1.5) resp. (1.6).Now, for any 2-category C with truncation τ ( C ), we can compose (6.10)for τ ( C ) with the natural projection C → ∆ o τ ( C ) = π ( C ) to obtain a specialfunctor(6.12) ξ : C → τ ( C ) . Then (6.10) and (6.12) enjoy the following universal properties.88 emma 6.11. Any special functor E : ∆ o I → E to some category E factorsuniquely through the functor (6.10) , and any special cofibration C ′ → ∆ o I isof the form C ′ ∼ = ξ ∗ C for a unique cofibration C → I .Proof. The functor ζ ([0]) of (2.1) for the cofibration ∆ o + I = ρ o ∗ ∆ o I → ∆ + gives a projection ∆ o + I → I Id , or in other words, a decomposition(6.13) ∆ o + I ∼ = a i ∈ I (∆ o + I ) i , a categorical version of (3.17), and then as in Lemma 3.2, for any i ∈ I , wehave an adjoint pair of functors(6.14) λ ( i ) = λ : i \ ξ ∆ o I ∼ = λ ∗ ρ ∗ (∆ o + I ) i → (∆ o + I ) i ,ρ ( i ) = a ∗ ρ ∗ : (∆ o + I ) i → i \ ξ ∆ o I, where a : λ ◦ ρ → id is the adjunction map. Therefore the subcategories(∆ o + I ) oi ⊂ (∆ o I ) o with the augmentations ρ o ( i ) = ρ ( i ) o give a framing forthe functor ξ o : (∆ o I ) o → I o in the sense of Lemma 1.14. Moreover, forany special functor E : ∆ o I → E , the opposite functor E o is locally con-stant on (∆ o + I ) i , thus constant since (∆ o + I ) i has an initial object. Thenby Lemma 1.14, ξ o ! E o exists, and the adjunction map ξ o ∗ ξ o ! E o → E o isan isomorphism, so that E o indeed factors through ξ o (and then E factorsthrough ξ ). For cofibrations, let C = ξ ∗ C ′ , and use the adjunctions (6.14)and equivalences (2.26) to check that the functor ξ ∗ C → C ′ of (2.22) is anequivalence. (cid:3) Corollary 6.12. Any special functor E : C → E from a -category C to acategory E factors uniquely through the functor (6.12) .Proof. By (2.24), E factors through ∆( E ), and then since C [0] is discrete, itfurther factors as C E ′ −−−−→ ∆ o E ξ −−−−→ E , where E ′ is cocartesian over ∆ o . Since ∆ o E → ∆ o is discrete, E ′ furtherfactors through π ( C ) = ∆ o τ ( C ), and we are done by Lemma 6.11. (cid:3) Remark 6.13. In terms of Definition 1.7, Corollary 6.12 can be rephrasedto say that (6.12) is a localization, and special maps are dense in χ ∗ Iso . Remark 6.14. An obvious counterpart of Corollary 6.12 for special cofi-brations is completely wrong. 89 .3 Cylinders and -functors. For any two 2-categories C , C ′ with C ′ bounded, bounded 2-functors from C ′ to C form a full subcategory inFun ∆ o ( C ′ , C ) that we denote by Fun ( C ′ , C ) ⊂ Fun ∆ o ( C ′ , C ). For any smallcategory I and 2-category C , we simplify notation by writing Fun ( I, C ) =Fun (∆ o I, C ). For any [ n ] ∈ ∆, we tautologically have Fun ([ n ] , C ) ∼ = C [ n ] .More generally, if are given a quiver Q : D o → Sets, a 2-category C definesa cofibration C ( Q ) → D o with fibers C ( Q ) i = C Q ( i )[ i ] , i = 0 , Lemma 6.15. For any quiver Q and -category C , we have a natural iden-tification (6.15) Fun ( P ( Q ) , C ) ∼ = Sec ♮ ( D o , C ( Q )) between -functors from the path category P ( Q ) to C and cocartesian sectionsof the cofibration C ( Q ) → D o . In particular, if we have a cocartesian square Q q : [1] → D o Sets of quiv-ers, so that P ( Q q ) : [1] → Cat is a cocartesian square of small categories,then by Lemma 6.15, the corresponding square of categories Fun ( P ( Q q ) , C )is cartesian. This of course includes the squares (6.2) but there are otheruseful examples. Proof. Let A ( Q ) = α o ∗ Q : ∆ oa → Sets be the right Kan extension withrespect to the functor (3.18), and let D o Q → D o , ∆ oa A ( X ) → ∆ oa be thediscrete cofibrations corresponding to Q and A ( Q ). Then (3.18) lifts to adiagram(6.16) D o Q α ( Q ) −−−−→ ∆ oa A ( Q ) β ( Q ) −−−−→ ∆ o P ( Q )of functors over ∆ o , and (2.25) provides an identification Sec ♮ ( D o , C ( Q )) ∼ =Fun ♮ D o ( D o Q, δ o ∗ C ). For any E ∈ Fun ♮ D o ( D o Q, δ o ∗ C ), the right Kan extension α ( Q ) ∗ E with respect to (6.16) can be computed by (the dual version of)(1.13), and since the cofibration ∆ oa A ( X ) → ∆ oa is discrete, it identifies theright comma-fibers of the functor α ( Q ) with those of α of (3.18). Then thelimits in the right-hand side of (1.13) reduce to iterated limits of standardfunctors of Remark 6.3, so that α ( Q ) ∗ exists and provides an equivalence ofcategories(6.17) Fun ♮ D o ( D o Q, δ o ∗ C ) ∼ = Fun ♮ ∆ oa (∆ oa A ( Q ) , β ∗ C )inverse to α ( Q ) ∗ . Now as in Remark 3.4, to compute P ( Q ) = β ! A ( Q ), onecan combine (2.16) and Example 2.8, and this provides an identification(6.18) ∆ o P ( Q ) ∼ = ar ± (∆) o♮ × ∆ oa ∆ oa A ( Q ) , t ( Q ) : ∆ o P ( Q ) → ∆ oa A ( Q )right-adjoint to β ( Q ). Then by (2.6), the relaive Kan extension β ∆ o ! providesan equivalence(6.20) Fun ♮ ∆ oa (∆ oa A ( Q ) , β ∗ C ) ∼ = Fun ♮ ∆ o (∆ o P ( Q ) , C ) = Fun (∆ o P ( Q ) , C ) , and to finish the proof, it remains to combine (6.20) and (6.17). (cid:3) Even in the simple case I = P ( Q ), a lax 2-functor from a category I toa 2-category C contains much more data than a 2-functor (for I = pt , this isconsidered below in Subsection 6.4). We do not attempt to prove any clas-sification results similar to Lemma 6.15. However, we do need one generalconstruction, namely, a 2-categorical version of the cylinder construction ofExample 2.1. Assume given categories I , I and a functor γ : I → I , andconsider the cylinder I = C ( γ ) with its cofibration π : I → [1]. Then wehave a functor ∆ o ( γ ) : ∆ o I → ∆ o I , and we can also consider the cylinder C (∆ o ( γ )). This is by definition a cofibration over [1] × ∆ o , (6.10) inducesa functor C (∆( γ )) → I × ∆ o cocartesian over [1] × ∆ o + , and by (2.24), thiscorresponds to a functor(6.21) α : C (∆ o ( γ )) → Id + ∗∗ ( I × ∆ o ) = ∆ o h I i cocartesian over [1] × ∆ o . Now assume given a 2-category C and two lax2-functors ϕ l : ∆ o I l → C , l = 0 , g : ϕ → ϕ ◦ γ .Then the triple h ϕ , ϕ , g i defines a single functor ϕ : C (∆ o ( γ )) → C , andwe have the following 2-categorical version of the cylinder construction. Lemma 6.16. The right Kan extension α ∗ ϕ : ∆ o h I i → C with respect to thefunctor (6.21) exists, and its restriction C ( γ, g ) : ∆ o I → C to ∆ o I ⊂ ∆ o h I i is a lax -functor.Proof. Extend ∆ o ( γ ) to a functor ∆ o ( γ ) > : ∆ o> I → ∆ o> I between aug-mented simplicial replacements, and consider the embedding ε : C (∆ o ( γ )) → C (∆ o ( γ ) > ). Then ε is left-closed in the sense of Example 1.15, so we havethe canonical extensions α > : C (∆ o ( γ ) > ) → ∆ o h I i > , ϕ > : C (∆ o ( γ ) > ) → C > of the functors α and ϕ , and α > ∼ = ε ∗ α , ϕ > ∼ = ε ∗ ϕ , so that ( α ∗ ϕ ) > ∼ = α > ∗ ϕ > ,where the right-hand side exists if and only if so does the left. Thus it suf-fices to prove that α > ∗ ϕ > exists and restricts to a lax 2-functor. We havethe embeddings σ l : I l → I , l = 0 , σ has a left-adjoint τ : I → I , and91he components α >l : ∆ o> I l → ∆ o h I i > , l = 0 , α > are givenby α >l = ∆ o ( σ l ) > . The functor α > has a left-adjoint α > † sending an object h [ n ] , i q i ∈ ∆ o h I i > to h [ n ] , τ ◦ i q i . Moreover, α > also has a left-adjoint α > † = µ of (3.19). Then for any object h [ n ] , i q i in ∆ o h I i > , we have a commutativesquare(6.22) h [ n ] , i q i −−−−→ α > ( h [ n ] , i q i ) y y α > ( h [ n ] , τ ◦ i q i ) −−−−→ α > ( h [ n ] , γ ◦ i q i )in the category ∆ o h I i > , where the arrows are the adjunction maps. Thisgives a functor V → C (∆ o ( γ ) > ) o / h [ n ] , i q i , and these functors form a framingfor α o in the sense of Lemma 1.14. If we now compute α ∗ using this framing,then the relevant limits over V reduce to colimits of half-standard functorsin the sense of Remark 6.3, thus exist, so that α > ∗ ϕ > exists. Moreover, by(6.22), it fits into a cartesian square(6.23) α > ∗ ϕ > −−−−→ ϕ > ◦ α > † y y g ϕ > ◦ α > † −−−−→ ϕ > ◦ ∆ o ( γ ) > ◦ α > † , and since α >l † , l = 0 , o ( γ ) > send anchor maps to anchor maps, C ( γ, g )is indeed a lax 2-functor. (cid:3) Remark 6.17. For an alternative construction of C ( γ, g ) that does not usethe category ∆ o h I i , decompose γ as(6.24) I σ −−−−→ I ζ −−−−→ I , where σ : I → I is the embedding and ζ is left-adjoint to the embedding I → I , and consider the functor ν : C (∆ o ( σ )) → C (∆ o ( γ )) over [1] given by ν = Id and ν = ∆ o ( ζ ). Then we have a commutative diagram(6.25) C (∆ o ( σ )) α ′ −−−−→ ∆ o I ν y y τ C (∆ o ( γ )) α −−−−→ ∆ o h I i , τ is the embedding, and α ′ is left-adjoint to the embedding ∆ o I = C (∆ o ( σ )) ⊂ C (∆ o ( σ )). Moreover, the framing (6.22) for the functor α liftsto a framing for the functor α ′ , and therefore the base change map C ( γ, g ) = τ ∗ α ∗ ϕ → α ′∗ ν ∗ ϕ is an isomorphism, so that its target can be used as a definition of C ( γ, g ). Let us now turn to non-symmetric monoidalstructures. In terms of Definition 6.1, these correspond to 2-categories witha single object. Definition 6.18. A unital monoidal structure on a category C is given bya 2-category B C such that B C [0] = pt is the point category, and B C [1] isequipped with an equivalence B C [1] ∼ = C . A lax monoidal structure on afunctor γ : C → C ′ between two categories equipped with unital monoidalstructures B C , B C ′ is given by a lax 2-functor Bγ : B C → B C ′ equippedwith an isomorphism Bγ [1] ∼ = γ . A lax monoidal structure Bγ is monoidal if it is a 2-functor.Explicitly, the composition − ◦ − in B C defines the tensor product − ⊗ − in C , and the identity object id pt ∈ C = B C ( pt , pt ) is the unit object 1 forthe tensor product. A lax monoidal structure on a functor γ is given by themaps (2.7) for the functor Bγ ; the essential ones are the maps(6.26) γ ( M ) ⊗ γ ( N ) → γ ( M ⊗ N ) , → γ (1)corresponding to m : [1] → [2] and the tautological projection [1] → [0]. Forany monoidal structure B C on a category C , the opposite 2-category B C ι also defines a monoidal structure on C (the product is the same but writtenin the opposite direction). For consistency, we denote C with this monoidalstructure by C ι . Example 6.19. For any 2-category C , and for any object c ∈ C [0] , the cat-egory C ( c, c ) carries a natural monoidal structure B C ( c, c ) ∼ = ε ( c ) ∗ C , where ε ( c ) : pt → C is the embedding onto c . Example 6.20. For any symmetric unital monoidal structure B ∞ C on acategory C in the sense of Definition 4.10, the pullback B C = Σ ∗ B ∞ C withrespect to the functor (3.16) is a unital monoidal structure on C in the senseof Definition 6.18, and we have a canonical identification B C ∼ = B C ι .93 xample 6.21. For any monoidal category C and bounded category I , thefunctor category Fun( I, C ) carries a natural pointwise monoidal structuregiven by B Fun( I, C ) ∼ = Fun( I, B C / ∆ o ). If C is symmetric, then this is thesame structure as in Example 4.11.Definition 6.1 and Definition 6.18 are pretty standard; however, it ismore common to use fibrations over ∆ rather than cofibrations over ∆ o .The two notions are equivalent – every fibration has its transpose cofibrationand vice versa – but it is the cofibrations that give the correct notion of alax 2-functor. In particular, lax monoidal functors pt → C are the samething as unital associative algebra objects in C , and it would be coalgebraswere we to use fibrations (the maps (6.26) would go in the other direction).Our definitions are also compatible with Definition 4.10 and Definition 4.14:a unital symmetric monoidal structure gives a unital monoidal structurevia pullback Σ ∗ with respect to the functor (3.16), and the same goes formonoidal and lax monoidal functors. Example 6.22. The category ∆ < is a unital monoidal category with re-spect to the concatenation product. The empty ordinal is the unit object,and [1] ∈ ∆ < is naturally an algebra object in ∆. The corresponding cofibra-tion B ∆ < → ∆ o is obtained by taking B ∆ < = ar ± (∆) o , with the projectionto ∆ o opposite to the fibration s of Example 2.12, and the Segal conditionis (3.14). In particular, the fiber B ∆ [1] is by definition the category ∆ o ± ,and this is canonically identified with ∆ < by (3.11). We also have the iden-tity section ∆ → ar ± (∆) of the projection s sending [ n ] to id : [ n ] → [ n ],and the opposite functor η : ∆ o → B ∆ is a lax monoidal functor pt → ∆corresponding to the algebra object 1 ∈ ∆ < . Example 6.23. Let ar p (∆) o ⊂ ar ± (∆) o be the full subcategory spannedby surjective arrows. Then the projection s o : ar p (∆) o → ∆ o is again acofibration. Its fiber ar p (∆) o [1] is natural identified with the category [1], and ar p (∆) o ∼ = B [1] defines a monoidal structure on [1] such that 0 ⊗ ⊗ ⊗ ⊗ η of Example 6.22 is universal in the followingsense. Denote by s, t : B ∆ = ar ± (∆) o → ∆ o the functors sending a bispecialarrow [ n ] → [ m ] in ∆ to its source [ n ] resp. its target [ m ], and for any 2-category C , let P ( C ) be the product(6.27) P ( C ) = C × t ∆ o B ∆ . P ( C ) ⊂ ar ( C ) is the full subcategory spanned by cocartesianlifting c → c ′ of bispecial arrows in ∆ o , and we have functors(6.28) s, t : P ( C ) → C sending c → c ′ to c ′ resp. c , and their common section(6.29) η : C → P ( C )sending c to id : c → c that is left-adjoint to t and right-adjoint to s . Interms of (6.27), the functor (6.29) is the product of id : C → C and the lax2-functor η of Example 6.22. Now, the projection s of (6.28) is a cofibrationthat turns P ( C ) into a 2-category, called the path -category of the 2-category C . Then (6.29) is a lax 2-functor, and any lax 2-functor γ : C → C ′ uniquelyfactors as(6.30) γ ∼ = P ( γ ) ◦ η, for a unique 2-functor P ( γ ) : P ( C ) → C ′ . Explicitly, we have P ( γ ) = η ∆ o ! γ ,where the relative Kan extension exists by Example 2.6 and is given by (2.6). Remark 6.24. Informally, the path 2-category P ( C ) has the same objectsas C , and morphisms in P ( C ) are free paths generated by morphisms in C ;this motivates the terminology. Note that if I is a small category, and P ( I )is the path category of the unverlying quiver, then P (∆ o I ) ∼ = ∆ o P ( I ).For any unital monoidal category C , the opposite category C o is alsounital monoidal; the corresponding cofibration B C o → ∆ o is given by B C o =( B C ) o ⊥ . In particular, this applies to ∆ Assume given a -category C , and a functor γ : ∆ o × [1] → C over ∆ o that is cocartesian over anchor maps. Then the right Kan extension w ∗ γ : B [1] → C with respect to (6.32) exists and defines a lax -functor,and the adjunction map w ∗ w ∗ γ → γ is an isomorphism. Conversely, forany lax -functor γ ′ : B [1] → C , the adjunction map γ ′ → w ∗ w ∗ γ ′ is anisomorphism.Proof. To compute w ∗ , let us choose a convenient framing of the oppositefunctor w o : ∆ × [1] o → ar p (∆). For any object in ar p (∆) representedby a surjective arrow g : [ n ] → [ m ], consider the corresponding diagram(3.4), and let v ( g ) : V ([ m ]) → ar p (∆) be the functor sending v ∈ V ([ m ])to the arrow [ n v ] → [0] (that is, to l o ([ n v ])). Then v ( g ) has a naturalaugmentation v ( g ) > : V ([ m ]) > → ar p (∆) sending o to g , with the maps l o ([ n v ]) → g induced by the map e g in (3.4). Extend v ( g ) > to a functor j ( g ) > : [1] × v ( g ) > → ar p (∆) equal to v ( g ) > on 1 × V ( g ) > and to r o ◦ π o ◦ v ( g ) > on 0 × v ( g ) > , with the adjunction map r o ◦ π o ◦ v ( g ) > → v ( g ) > , and notethat [1] × v ( g ) > has the largest element 1 × o , so that [1] × v ( g ) > = J ( g ) > for the partially ordered set J ( g ) = ([1] × v ( g ) > ) \ { × o } , and j ( g ) > isan augmentation of a functor j ( g ) : J ( g ) → ar p (∆). This functor j ( g )canonically factors through w o , and the induced functor J ( g ) → ar p (∆) / w o f is a left-admissible full embedding.The collection J ( g ) is our framing, and then for any γ : ∆ o × [1] → C with components γ , γ : ∆ o → C , the expected object w ∗ γ ( g ) ∼ = lim J ( g ) γ w ∗ γ ( g ) −−−−→ γ ([ n ]) y y e og ! Q v γ ([ n v ]) −−−−→ Q v γ ([ n v ])in the category C . This can be reinterpreted as opposite to an iteratedcolimit of half-standard functors V o → C of Remark 6.3, as in Lemma 6.16,so that if γ and γ are lax 2-functors, the limit exists, and then so does theKan extension w ∗ γ . The equivalence w ∗ w ∗ γ ∼ = γ is then obvious from thecartesian square (6.33), and since any lax 2-functor γ ′ : B [1] = ar p (∆) o → C preserves the limits of standard functors, we also have γ ′ ∼ = w ∗ w ∗ γ ′ . (cid:3) Remark 6.26. In the situation of Lemma 6.25, one can also separate γ intolax 2-functors γ , γ : ∆ o → C equipped with a map g : γ → γ , and considerthe 2-cylinder C ( id , g ) : ∆ o [1] → C provided by Lemma 6.16. The differencebetween C ( id , g ) and the right Kan extension w ∗ γ of Lemma 6.25 is that∆ o [1] does not correspond to a monoidal category: it has two objects. Since C ( id , g ) and w ∗ γ enjoy essentially the same universal property, we have e ∗ w ∗ γ ∼ = C ( id , g ), where e : ∆ o [1] → B [1] is the 2-cylinder of the map l → r used to define w , but of course e is not an equivalence.For a useful application of the same combinatorics to 2-categories ratherthen lax 2-functors, assume given a 2-functor γ : C → C between some2-categories C , C , and note that the cylinder C ( γ ) is then a cofibrationover ∆ o × [1] that restricts to C l over ∆ o × l , l = 0 , Definition 6.27. The wreath product C ≀ γ C of the 2-categories C , C withrespect to the 2-functor γ is the cofibration C ≀ γ C = w ∗ C ( γ )over B [1] = ar p (∆) o , where w is the functor (6.32).Explicitly, the wreath product C ≀ γ C can be computed by the sameframing that gives (6.33). This shows that the fiber ( C ≀ γ C ) g over somesurjective g : [ n ] → [ m ] is the category of triples h c , c , α i of a 2-functor c : [ n ] → C , a 2-functor c : [ n ] g → C , and an isomorphism α : e ∗ g c ∼ = c .In particular, C ≀ γ C is a 2-category, and we have C ≀ γ C ∼ = C ≀ e γ γ ∗ C , where e γ is the dense component of the decomposition (6.6). We will simplify notationby writing C ≀ γ C = C ≀ C when γ is clear from the context.97 xample 6.28. If C = pt is the point 2-category, then the only non-trivial part of a triple h c , c , α i describing an object in C ≀ pt is c . Thenthe projection t : ar p (∆) → ∆ of (1.6) induces a projection C ≀ pt → ∆ o whose fibers are given by(6.34) ( C ≀ pt ) [ m ] ∼ = C m +10 , [ m ] ∈ ∆ o . This motivates our terminology. If moreover C = ∆ o e ( S ) is the simplicialreplacement of a category e ( S ), as in Example 6.8, then we have(6.35) ∆ o e ( S ) ≀ pt ∼ = ∆ o e ( S ) × B [1] . Slightly more generally, for any 2-category C and functor S : C [0] → Sets,as in (6.7), we have the 2-functor π : C [ S ] → C , and (6.35) induces anidentification(6.36) C [ S ] ≀ π C ∼ = C × B [1] . On the other hand, if C is arbitrary but C is discrete, we have(6.37) C ≀ γ C ∼ = γ ∗ C × t o ∆ o B [1] , where t o : B [1] = ar p (∆) o → ∆ o is induced by (1.6), and γ ∗ C is as in (6.5).The wreath product construction is obviously functorial with respect to2-functors: if we are given another 2-functor γ ′ : C ′ → C ′ , and 2-functors ϕ : C → C ′ , ϕ : C → C ′ equipped with an isomorphism α : γ ′ ◦ ϕ ∼ = ϕ ◦ γ ,then we have a functor C ( ϕ ) : C ( γ ) → C ( γ ′ ) cocartesian over ∆ o × [1], andit induces a 2-functor(6.38) ϕ ≀ α ϕ = w ∗ C ( ϕ ) : C ≀ γ C → C ≀ γ ′ C , cocartesian over B [1], where we will again drop α from notation when itis clear from the context. To extend this to lax 2-functors, consider theuniversal situation: take the path 2-categories P ( C ), P ( C ), with the functor P ( C ) → P ( C ) induced by γ . Then s of (6.28) induces a 2-functor(6.39) s ≀ s : P ( C ) ≀ P ( C ) → C ≀ C , again cocartesian over B [1]. Lemma 6.29. The functor (6.39) admits a fully faithful right-adjoint lax -functor η ≀ η : C ≀ C → P ( C ) ≀ P ( C ) over B [1] . roof. By Lemma 2.10 (ii), it suffices to check that for any f ∈ B [1], thefiber ( s ≀ s ) f : ( P ( C ) ≀ P ( C )) f → ( C ≀ C ) f of the cocartesian functor (6.39)admits a right-adjoint. But by (6.33), its source resp. its target is a productof categories of the form P ( C ) [ n ] , P ( C ) [ m ] resp. ( C ) [ n ] , ( C ) [ m ] for various[ n ] , [ m ] ∈ ∆ o , and ( s ≀ s ) f is the product of the fibers s [ n ] of the functors(6.28). These have fully faithful right-adjoints induced by (6.29). (cid:3) Now if we have 2-functors γ : C → C , γ ′ : C ′ → C ′ , and lax 2-functors ϕ : C → C ′ , ϕ : C → C ′ equipped with an isomorphism γ ′ ◦ ϕ ∼ = ϕ ◦ γ ,we can define ϕ ≀ ϕ by(6.40) ϕ ≀ ϕ = ( P ( ϕ ) ≀ P ( ϕ )) ◦ ( η ≀ η ) : C ≀ C → C ≀ C , where P ( ϕ ), P ( ϕ ) are as in (6.30), and η ≀ η is provided by Lemma 6.29.This is a functor over B [1]. If ϕ , ϕ are actual 2-functors, then we have( s ≀ s ) ◦ ( η ≀ η ) ∼ = Id since η ≀ η is fully faithful, and (6.40) agrees with (6.38).In addition to that, if we have a lax 2-functor γ : C → C , we can define thewreath product C ≀ γ C by the cartesian square(6.41) C ≀ γ C −−−−→ P ( C ) ≀ P ( γ ) C y y id ≀ τ C ≀ pt η ≀ id −−−−→ P ( C ) ≀ pt , where τ : C → pt is the tautological projection. This again agrees withDefinition 6.27 when γ is a 2-functor. In all cases, we have a natural pro-jection(6.42) C ≀ C γ ≀ id −−−−→ C ≀ id C ∼ = C × B [1] −−−−→ C , where the identification in the middle is (6.36). It is a lax 2-functor, and a2-functor if so is γ . Now assume given a category C equipped with a unitalmonoidal structure B C . Definition 6.30. A module over C is a category M equipped with a functor µ : M → B C such that the composition M → B C → ∆ o is a cofibration,with the induced cofibration ρ o ∗ M → ∆ o + , and the functor(6.43) ζ ([0]) × ρ o ∗ µ : ρ o ∗ M → M × ρ o ∗ B C , M = M [0] is an equivalence of categories. A morphism between two C -modules M , M ′ is a functor α : M ′ → M over B C cocartesian over all special maps in ∆ o .99xplicitly, (6.43) provides identifications M [ n ] ∼ = M × B C [ n ] ∼ = M × C n ,[ n ] ∈ ∆, and prescribes the transition functors f o ! : M [ n ] → M [ n ′ ] for allspecial maps f : [ n ′ ] → [ n ] in ∆. The essential part of the structure is thefunctor(6.44) m = t o ! : M × C → M corresponding to the antispecial map t : [0] → [1]. This functor defines anaction of C on M , and the rest of the structure encodes the usual associativityand unitality constraints for this action. A morphism α is then given by afunctor α [0] : M ′ = M ′ [0] → M and a map(6.45) α [0] ◦ m ′ → m ◦ ( α [0] × id ) . In particular, α can be non-trivial even if α [0] = id . Example 6.31. For any unital monoidal category C , κ o ∗ C with the pro-jection a o ! : κ o ∗ C → C is a C -module; the corresponding action (6.44) isthe action of C on itself by left products. For any category E , the product E × B C is a (trivial) C -module; the corresponding action (6.44) is the projec-tion onto the first factor. For any C -module M with the action map (6.44),and unital monoidal functor γ : C ′ → C from a unital monoidal category C ’,the pullback γ ∗ M → B C ′ is a C ′ -module, with the action map m ◦ ( id × γ ).For any C -module M , the equivalence (6.43) immediately implies that ρ o ∗ µ is both a fibration and a cofibration, but µ itself is neither: in general,it is only a precofibration. To understand the structure of the category M better, it is useful to consider the category ∆ q of (3.7), with the functor ν † :∆ → ∆ q and its two adjoints ν q , ν ⊥ : ∆ q → ∆. If we denote B q C = ν o ∗ q B C ,with the induced fibration ν q : B q C → B C , then ν † and ν ⊥ induce functors ν † : B C → B q C , ν ⊥ : B q C → B C , and we can consider the precofibration(6.46) µ q : M q = M × ν ⊥ B C B q C → B q C . Then the composition ν q ◦ µ q : M q → B C is actually a fibration. The fiber ofthis fibration over some h [ n ] , c q i ∈ B C , c q ∈ C n is the category M ( h [ n ] , c q i )cofibered over [ n ] o , with all fibers identified with M , and transition functors m ( − × c i ) : M → M , i = 1 , . . . , n . For any map f : [ n ′ ] → [ n ] in ∆,we have f ∗ M ( h [ n ] , c q i ) ∼ = M ( h [ n ′ ] , f ∗ c q i , so that M ( h [ n ] , c i ) is covariantlyfunctorial with respect to [ n ] ∈ ∆, and (2.3) immediately shows that it isalso contravariantly functorial with respect to c q .100ssume given a C -module M with bounded M = M [0] , and some othercategory E . Then the action functor (6.44) induces a functor(6.47) m : Fun( M, E ) × C → Fun( M, E ) , m ( F × c )( c ′ ) = F ( c × c ′ ) , and it turns out that this extends to a module over the opposite monoidalcategory C ι . To construct it, consider the precofibration (6.46), take the cor-responding precofibration µ ( E ) : Fun( M q /B q C , E ) → B q C , and note that thecomposition ν q ◦ µ ( E ) : Fun( M q /B q C , E ) → B C is a cofibration, with fibersFun( M ( h [ n ] , c i ) / [ n ] , E ). Therefore we can consider the transpose fibrationFun( M /B q C , E ) ⊥ → B C , and µ ( E ) induces a functor(6.48) ι ∗ µ ( E ) ⊥ : ι ∗ Fun( M /B q C , E ) ⊥ → ι ∗ ( B q C ) ⊥ ∼ = B q C ι , where ( B q C ) ⊥ is taken with respect to the fibration ν q : B q → C . Then(6.49) Fun ⊗ ( M , E ) = ν ∗† ι ∗ Fun( M /B q C , E ) ⊥ , with the induced functor Fun ⊗ ( M , E ) → ν ∗† B q C ι ∼ = B C ι , is a C ι -module, withFun ⊗ ( M , E ) [0] ∼ = Fun( M, E ) and the action (6.47), while (6.48) is identifiedwith the corresponding precofibration Fun ⊗ ( M , E ) q → B q C ι of (6.46).As an application of this construction, let Γ + be the category of pointedfinite sets, with the unital monoidal structure given by smash product, letit act on itself, and let E be a half-additive category in the sense of Sub-section 4.3. Then the functor (4.12) induced a fully faithful embedding m † : E → Fun(Γ + , E ), and for any [ n ] ∈ ∆, we have a fully faithful em-bedding m † × id : E × Γ n + → Fun ⊗ (Γ + , E ) [ n ] ∼ = Fun(Γ + , E ) × Γ n + . If welet E ⊗ ⊂ Fun ⊗ (Γ + , E ) be the full subcategory spanned by the essential im-ages of these embedding, with the induced projection E ⊗ → B Γ + , then byLemma 2.11 (i), E ⊗ becomes a module over Γ + ∼ = Γ ι + in the sense of Defini-tion 6.30. The corresponding action functor (6.44) is the functor (4.12), andthe embedding E ⊗ → Fun ⊗ (Γ + , E ) is a morphism of Γ + -modules. Moreover,assume that E has kernels , in the sense that for any map f : e ′ → e , thereexists Ker( f ) = o × e e ′ , where o ∈ E is the initial terminal object. Thenessentially as in Lemma 4.9, m † admits a right-adjoint Fun(Γ + , E ) → E sending E : Γ + → E to the kernel of the map E ( pt + ) → E ( o ), and then theright-adjoint(6.50) Fun ⊗ (Γ + , E ) → E ⊗ to the embedding E ⊗ → Fun ⊗ (Γ + , E ) provided by Lemma 2.11 (i) is also amorphism of Γ + -modules. 101ote that the category Fun(Γ + , E ) is also half-additive, so that effec-tively, it has two structures of a Γ + -module: Fun ⊗ (Γ + , E ) on one hand, andFun ⊠ (Γ + , E ) = Fun(Γ + , E ) ⊗ of (6.50) on the other hand (informally, Γ + canact on Fun(Γ + , E ) either via Γ + for via E ). To relate the two, let Γ + act onΓ + × Γ + via the left factor. Then the product functor m : Γ + × Γ + → Γ + gives rise to a morphism µ of Γ + -modules, and we have the morphism(6.51) Fun ⊗ (Γ + , E ) m ∗ −−−−→ Fun ⊗ (Γ + × Γ + , E ) −−−−→ Fun ⊠ (Γ + , E ) , where the first arrow is induced by m , and the second one is the morphism(6.50) with the identification Fun ⊗ (Γ + × Γ + , E ) ∼ = Fun ⊗ (Γ + , Fun(Γ + , E )).More generally, if in addition to E we also have a bounded category I ,then the functor category Fun( I, Γ + ) equipped with the pointwise productof Example 6.21 acts on Γ + × I byΓ + × I × Fun( I, Γ + ) → Γ + × I, S + × i × F S + ∧ F ( i ) × i, and the same construction equips the category Fun( I × Γ + , E ) with twostructures of a module over Fun( I, Γ + ), while (6.51) induced a morphism(6.52) Fun ⊗ ( I × Γ + , E ) → Fun ⊠ ( I × Γ + , E )between the corresponding Fun( I, Γ + )-modules. For any 2-category C and two objects c, c ′ ∈C [0] , an adjoint pair of maps between c and c ′ is a quadruple h f, f ∨ , a, a ∨ i , f ∈ C ( c, c ′ ), f ∨ ∈ C ( c ′ , c ), a : id c → f ∨ ◦ f , a ∨ : f ◦ f ∨ → id c ′ , subject to theusual relations(7.1) ( a ∨ ◦ id f ) ◦ ( id f ◦ a ) = id f , ( id f ∨ ◦ a ∨ ) ◦ ( a ◦ id f ∨ ) = id f ∨ . For any c, c ′ ∈ C [0] , adjoint pairs of maps between c and c ′ and isomorphismsbetween them form a category that we denote A dj ( C )( c, c ′ ), and the forget-ful functor from A dj ( C )( c, c ′ ) to C ( c, c ′ ) Iso sending h f, f ∨ , a, a ∨ i to f is fullyfaithful (for the 2-category of small categories, this is the standard unique-ness of adjoints, and the same proof works for a general 2-category C ). Wesay that f ∈ C ( c, c ′ ) is reflexive if it extends to an adjoint pair, and wenote that reflexivity is closed under compositions (again, the proof for smallcategories works in the general case, or see below in Subsection 7.5). Thus102e actually have the adjunction -category A dj ( C ) with the same objects as C and A dj ( − , − ) as categories of morphisms, and we have a fully faithful2-functor(7.2) A dj ( C ) → C ♮ that is an identity over [0]. If A dj ( C ) is bounded, we denote its truncationby Adj( C ) = τ ( A dj ( C )). A reflexive morphism f ∈ C ( c, c ′ ) is an equivalence if there exists an adjoint pair h f, f ∨ , a, a ∨ i with invertible a and a ∨ .To understand adjunction in the general 2-categorical context, it is usefulto look at the universal situation. Let adj be the 2-category with two objects0, 1, and categories of morphisms(7.3) adj (0 , 0) = ∆ < , adj (0 , 1) = ∆ − , adj (1 , 0) = ∆ + , adj (1 , 1) = ∆ ± , with compositions∆ < × ∆ − → ∆ − , ∆ + × ∆ < → ∆ + , ∆ < × ∆ < → ∆ < given by the concatenation product − ◦ − , and compositions∆ ± × ∆ + → ∆ + , ∆ − × ∆ ± → ∆ − , ∆ ± × ∆ ± → ∆ ± given by the reduced concatenation product − ∗ − . Then we have twomorphisms f : 0 → f ∨ : 1 → adj corresponding to the initial objectsin ∆ + , ∆ − , and f ∨ ◦ f ∼ = [1] ∈ ∆ < , f ◦ f ∨ ∼ = [1] ∈ ∆ C × 1, and an equivalence is a2-functor(7.9) C × eq → C ′ . A natural transformation is reflexive if it extends to an adjoint pair, and anequivalence if it extends to an equivalence. Example 7.2. Assume given a 2-category C and a functor S : C [0] → Sets,and let C [ S ] be the corresponding 2-category of (6.7), with its 2-functor π : C [ S ] → C . Moreover, assume that S ( c ) is non-empty for any c ∈ C [0] ,so that π admits a section σ : C = C [ pt ] → C [ S ]. Then p ◦ σ ∼ = Id , and σ ◦ π : C [ S ] → C [ S ] is equivalent to Id in the sense of (7.9). To construct anequivalence (7.9) between σ ◦ π and id , note that C [ S ] × eq ∼ = C [ S × { , } ],and consider the map S × { , } → S equal to id on S × { } and to thecompositon S → pt → S on S × { } .As we see from the explicit description of the 2-category adj in terms of(7.5) and (7.6), we actually have a full embedding(7.10) adj → P ( eq ) ∼ = ar ± ( eq )into the path 2-category of eq = ∆ o e ( { , } ). The projection ν : adj → eq isinduced by the cofibration s : P ( eq ) → eq of (6.28), thus has a right-adjoint(7.11) η : eq → adj ⊂ P ( eq )105nduced by (6.29). However, η in turn has a right-adjoint t : adj ⊂ P ( eq ) → eq → ∆ o induced by (6.28). For any 2-category C , we denote C{ adj } = C × t ∆ o adj ⊂ P ( C ( eq )), and we note that the functors (6.28) and (6.29)induce functors(7.12) s, t : C{ adj } → C{ eq } , η : C{ eq } → C{ adj } , where we denote C{ eq } = C × eq = C [ { , } ] for consistency. We furthernote that s is a cofibration whose composition with the cofibration eq → ∆ o turns C{ adj } into a 2-category. We then define a coadjoint pair of functorsfrom C to some 2-category C ′ as a 2-functor(7.13) γ : C{ adj } → C ′ . If C is discrete — that is, C ∼ = C [0] × ∆ o — then C{ adj } ∼ = C [0] × adj ∼ = C × adj ,and a coadjoint pair (7.13) is the same thing as an adjoint pair (7.8). Thisis useful since coadjoint pairs are much easier to construct.Namely, let ι l : ∆ o> → eq > , l = 0 , pt → e ( { , } ) onto l , and note that the embeddings ι , ι admitleft-adjoint functors(7.14) s , s : eq > → ∆ o> sending an injective arrow a : [ n ] → [ n ] to [ n ] resp. [ n ] (we have to pass tothe augmented categories since [ n ] or [ n ] might be empty). GeneralizingLemma 3.8, say that a map f in eq is l -special , l = 0 , s l ( f ) is invertible.For any 2-category C equipped with a 2-functor C → eq , say that a map f in C is l -special if it is a cocartesian lifting on an l -special map in eq , andsay that a lax 2-functor C → C ′ to some C ′ is l -special if it sends l -specialmaps in C to maps in C ′ cocartesian over ∆ o . Note that for any C / eq , theembeddings ι ∗ l C > → C > , l = 0 , s l : C > → ι ∗ l C > , l = 0 , , and s l inverts l -special maps. In these terms, for any 2-category C , C{ adj } ⊂ P ( C{ eq } ) is the full subcategory spanned by 1-special maps with respect tothe projection C{ eq } → eq . Remark 7.3. More generally, for any 2-category C equipped with a 2-functor π : C → eq , say that a map f in C is l -anchor , l = 0 , 1, if it iscocartesian over eq , and s l ( π ( f )) is an anchor map. Then any l -special lax2-functor γ : C → C ′ also sends l -anchor maps to maps cocartesian over ∆ o> ,and so does s l . 106 emma 7.4. For any -category C , the functor η of (7.12) is a -special lax -functor with respect to the projection C{ eq } → eq . Moreover, any -speciallax -functor γ : C{ eq } → C ′ to some -category C factors as (7.16) C{ eq } η −−−−→ C{ adj } γ ′ −−−−→ C ′ , where γ ′ is a -functor, and the factorization is unique up to a unique iso-morphism.Proof. As in Lemma 3.8, 0-special and 1-special maps in eq o = ∆ e ( { , } )form a factorization system — namely, any map f : h [ m ] , f ∗ e q i → h [ n ] , e q i in ∆ e ( { , } ) can be uniquely factored as(7.17) h [ m ] , f ∗ e q i f −−−−→ h [ m ] f , f ∗ e q i f −−−−→ hh [ n ] , e q i with 0-special f o and 1-special f o (indeed, by virtue of (3.5), it sufficesto construct (7.17) when [ n ] = [0], and then either f = id , f = f , orthe other way around, depending on e q (0) ∈ { , } ). Moreover, we alsohave a factorization system given by bispecial and anchor maps, and if f isbispecial, then so are its components f , f . Then we have adj ∼ = ar ± ( eq ),where 1 ± is the class of bispecial 1-special maps, and the embedding (7.10)admits a left-adjoint functor(7.18) l : P ( eq ) → adj ⊂ P ( eq )sending an arrow f to the 1-special component f of its decomposition (7.17).For any 2-category C , (7.18) induces a functor l : P ( C{ eq } ) → C{ adj } left-adjoint to the embedding C{ adj } ⊂ P ( C{ eq } ).Now, any lax 2-functor γ : C{ eq } → C ′ has the canonical decomposition(6.30), and γ is 0-special if and only if P ( γ ) inverts maps in P (0) = ( s × t ) ∗ ( Iso × C{ eq } . If γ = η is theembedding (7.16), then P ( γ ) = l is the functor (7.18) that does invert mapsin P (0). Moreover, (7.18) is a localization, with the adjunction map c → l ( c )in P (0) for any c ∈ P ( C{ eq } ), so that for any γ , P ( γ ) inverts maps in P (0)if and only if it factors through l , and the factorization is unique. (cid:3) By Lemma 7.4, the study of coad-joint pairs reduces in large part to the combinatorics of the 2-category eq .Let us prove several results in this direction. First, consider the functor s : eq > → ∆ o> of (7.15) for the point 2-category C = pt . Recall thatobjects in eq > can also be described by injective maps a : [ n ] → [ n ] in ∆ < ,107nd let eq ⊂ eq ⊂ eq > be the full subcategory spanned by bispecial a (orequivalently, by h [ n ] , e q i ∈ ∆ o e ( { , } ) = eq such that e q (0) = e q ( n ) = 1). Lemma 7.5. The functor s > : eq > → ∆ o> is a cofibration whose everyfiber has an initial object, and eq ⊂ eq > is a subcofibration that containsall these initial objects.Proof. Recall that 0-special and 1-special maps in eq > form a factorizationsystem, in either order (in one of the orders this is (7.17)). Then by Exam-ple 2.12, we have a cofibration t : ar ( eq < ) → eq > , where ar ( eq > ) ⊂ ar ( eq > )is spanned by 0-special arrows, and as in Lemma 3.8, we observe that ι ∗ ar ( eq > ) ∼ = eq > , with the equivalence sending x ∈ eq > to the adjunc-tion arrow ι ( s ( x )) → x . This equivalence identifies s with the cofibration ι ∗ ( t ), so that s is a cofibration. Its fiber eq > [ m ] for some [ m ] ∈ ∆ o> is thenequivalent to the category of injective maps [ m ] → [ n ], with [ m ] correspond-ing to [ n ]. The functor (6.31) identifies the opposite to this category withthe left comma-fiber ∆ < / [ m + 1], and by (3.5), this has a terminal object[0] m +1 . Explicitly, the corresponding initial object in eq > [ m ] is h [2 m + 2] , e q i ,where e q is given by(7.19) e l = l + 1 mod 2 , l ∈ [2 m + 2] = { , . . . , m + 2 } , so that it manifestly lies in eq for any m . Finally, by construction, a map f : x → y in eq > is cocartesian with respect to s if and only if it is 1-special,and then if its source x lies in eq , so does its target, so that eq ⊂ eq > is asubcofibration by Lemma 2.11 (i). (cid:3) Remark 7.6. According to the universal property of the 2-category adj , thedense subcategory adj ♮ ⊂ adj defined by maps cocartesian over ∆ o shouldbe the category freely generated by the morphisms f , f ∨ but without theadjuntion maps a , a ∨ — in other words, adj ∼ = ∆ o [2] Λ ∼ = ∆ o P ([2] λ ) is thesimplicial replacement of the path category of the wheel quiver [2] λ with twovertices 0, 1 and two arrows f : 0 → f ∨ : 1 → 0. To see this explicitly,one can use (7.19). Namely, say that an object h [ n ] , e q i ∈ eq is alternating if e l +1 = e l + 1 mod 2, 0 ≤ l < n . Then alternating objects correspond topath in the quiver [2] λ , an object h [ n ] , e q i of the form (7.19) is alternating,and for any anchor map a : [ m ] → [ n ], so is the induced object h [ m ] , a ∗ e q i .Now, let a eq of the form a ◦ f with 0-special f and anchor a . Then we have a factorization system h a , ±i on eq , and adj ∼ = ar ± ( eq ), so that by Example 2.13, we have(7.20) adj ♮ ∼ = adj ∩ P ( eq ) t ∗ a ⊂ P ( eq ) , l : P ( eq ) t ∗ a → adj ♮ left-adjoint to thefull embedding adj ♮ → P ( eq ) t ∗ a . However, Lemma 7.5 immediately showsthat l itself has a left-adjoint p : adj ♮ → P ( eq ) t ∗ a . It is automatically fullyfaithful, and by (7.19), its essential image is spanned by bispecial arrows in eq with alternating target. This is exactly ∆ o P ([2] λ ), in the form (6.18),and (6.19) provides a functor(7.21) t : adj ♮ → ∆ oa A ([2] λ ) , where ∆ oa A ([2] λ ) ⊂ eq a is the full subcategory spanned by alternating ob-jects.Next, consider the wreath product eq ≀ pt of Definition 6.27. We have theprojection t o : eq ≀ pt → ∆ o of Example 6.28, with fibers (6.34). By abuse ofnotation, let eq ≀ pt ⊂ eq ≀ pt be the full subcategory spanned by eq m +11 ⊂ eq m +1 ∼ = ( eq ≀ pt ) [ m ] , [ m ] ∈ ∆ o . We also have eq ≀ pt ∼ = eq × B [1] by (6.35),and if we further abuse notation by denoting eq > ≀ pt = eq > × s o ar (∆ < ) o ,then we have full embeddings eq ≀ pt ⊂ eq ≀ pt ⊂ eq > ≀ pt . The projections(1.6) and (1.5) induce a cofibration s o : eq > ≀ pt → eq > and a fibration t o : eq > ≀ pt → ∆ o> with fibers ( eq > ≀ pt ) [ m ] ∼ = eq > ( m +1) , [ m ] ∈ ∆ o . We alsohave an embedding ι × id : ar (∆ < ) o → eq > ≀ pt cartesian over ∆ o , and thefiber over each [ m ] ∈ ∆ o> , it has a left-adjoint(7.22) ψ ′ m ∼ = s m +10 : ( eq > ≀ pt ) [ m ] ∼ = ( eq > ) m +1 → ar (∆ < ) o [ m ] ∼ = (∆ o> ) m +1 , where the last identification is (3.5). By Lemma 2.10 (ii), ι × id then has aleft-adjoint ψ ′ : eq > × pt → ar (∆ < ) o over ∆ o> that fits into a commutativesquare(7.23) eq > ≀ pt s o −−−−→ eq >ψ ′ y y s ar (∆ < ) o s o −−−−→ ∆ o> . Restricting ψ ′ to eq ≀ pt , we obtain functors(7.24) eq ≀ pt ψ −−−−→ ar (∆ < ) o s o −−−−→ ∆ o> , and we denote their composition by χ . Lemma 7.7. The composition χ : eq ≀ pt → ∆ o> of the functors (7.24) isa cofibration, the functor ψ is cocartesian over ∆ o> , and each of its fibers ψ [ n ] , [ n ] ∈ ∆ o> has a fully faithful left-adjoint ψ † [ n ] . roof. First observe that all the statements hold if we replace eq ≀ pt with eq > ≀ pt , and ψ , χ with ψ ′ , χ ′ = s o ◦ ψ ′ . Indeed, s in (7.23) is a cofibrationby Lemma 7.5, therefore so is χ ′ ∼ = s ◦ s o . A map f in eq > ≀ pt cocartesianwith respect to χ must also be cocartesian with respect to s o , this happensif and only if t o ( f ) is invertible, and since ψ ′ commutes with t o , t o ( ψ ′ ( f )) isthen invertible, so that ψ ′ ( f ) is cocartesian with respect to s o . Dually, s o inverts maps cartesian with respect to t o , so that for any [ n ] ∈ ∆ o> , both t o : ( eq > ≀ pt ) [ n ] → ∆ o> and t o : ar (∆ < ) o [ n ] ∼ = ∆ o> / [ n ] → ∆ o> are fibrations,and since the latter is discrete, ψ ′ [ n ] is also a fibration. But each of its fibershas an initial object by Lemma 7.5, so it has a fully faithful left-adjoint byLemma 2.10 (ii).Now note that since any map f in eq > ≀ pt cocartesian over ∆ o> is in-verted by t o , it must lie entirely in one of the fibers (7.22), and then byLemma 7.5, if its source is in eq ≀ pt ⊂ eq > ≀ pt , then so is its target.Therefore eq ≀ pt ⊂ eq > ≀ pt is a subcofibration, and ψ is cocartesian over∆ o> . To finish the proof, it remains to observe that by (7.19), all the ini-tial objects in the fibers of the fibrations ψ [ n ] also lie in eq ≀ pt , and applyExample 1.5. (cid:3) Note that by Lemma 2.10 (ii),(iv), Lemma 7.7 immediately implies that ψ is a localization in the sense of Definition 1.7. Since s o in (7.24) has afully faithful right-adjoint η o induced by (7.11), it is also a localization byExample 1.9, and then so is χ = s o ∼ = ψ . Thus by Example 1.12, for anyfunctor E : eq ≀ pt → E to some category E that inverts maps in χ ∗ Iso , χ ! E and χ ∗ E exist, and we have χ ! E ∼ = χ ∗ E and E ∼ = χ ∗ χ ! E ∼ = χ ∗ χ ! E . Wealso have χ ! E ∼ = s o ! ψ ! E ∼ = η o ∗ ψ ! E , or the same with ψ ∗ E instead of ψ ! E ,but neither ψ ! E not ψ ∗ E cannot be expressed as a pullback since ψ doesnot have an adjoint. For any map f : [ m ] → [ n ] in ∆ o> , we have the map ψ † [ n ] ◦ f ! → f ! ◦ ψ † [ m ] adjoint to the map (2.7) for ψ , but it goes in the wrongdirection, so that ψ † [ n ] do not form a functor over ∆ o> . On objects, we stillhave canonical isomorphisms(7.25) ψ ! E ([ n ]) ∼ = E ( ψ † [ n ] ( η o ([ n ]))) , [ n ] ∈ ∆ o> . Explicitly, we can think of objects in eq ≀ pt as triples h [ n ] , [ n ] , p i of anobject [ n ] ∈ ∆ < , a subset [ n ] ⊂ [ n ], and a surjective map p : [ n ] → [ m ]to some [ m ] ∈ ∆ < . Then for any [ n ] ∈ ∆ o> , η o ([ n ]) is the identity arrow id : [ n ] → [ n ], and by (7.19), ψ † [ n ] sends it to the triple(7.26) β ([ n ]) = h [ n ] × [2] , [ n ] × [1] , p i , n ] × [2] and [ n ] × [1] are equipped with the lexicographical order,the embedding [ n ] × [1] ⊂ [ n ] × [2] is id × a , where a : [1] → [2] is the uniquebispecial embeddding, and p is the projection p : [ n ] × [2] → [ n ]. However,if one tries to write down explicitly ψ ! ( E )( f ) for some map f : [ n ] → [ m ],one ends up with a zigzag of length 3. Let us now use the combinatorics of Sub-section 7.3 to prove a useful general result on coadjoint pairs. First, assumegiven a 2-category C , let γ : C [0] { eq } → C [0] → C be the composition ofthe projection C [0] { eq } = C [0] × eq → C [0] and the tautological embedding C [0] → C , and denote W ( C ) = C [0] { eq } ≀ γ C . We then have γ ∗ C ∼ = C{ eq } , and(6.35) together with (6.37) provide an identification(7.27) W ( C ) ∼ = C × t o B [1] s o × eq . In particular, we have a cofibration W ( C ) → eq × s o B [1] = eq ≀ pt , and wecan define a full subcategory W ( C ) ⊂ W ( C ) by(7.28) W ( C ) = W ( C ) × eq ≀ pt eq ≀ pt ∼ = C × t o eq ≀ pt . The functors (7.24) then induce functors(7.29) W ( C ) id × ψ −−−−→ ar ‡ ( C > ) t −−−−→ C > , where ar ‡ ( C > ) ∼ = C > × t o ar (∆ > ) o is the subcategory in ar ( C > ) spanned byarrows cartesian with respect to the fibration (1.6), and t is the cofibration(1.5). Denote the composition of the functors (7.29) by(7.30) χ : W ( C ) → C > . Note that W ( C ) is a 2-category equipped with a projection to eq , so it makessense to speak of 0-special maps in W ( C ). By abuse of terminology, say thata map in W ( C ) is 0-special if it is 0-special as a map in W ( C ). Corollary 7.8. The functor (7.30) is a localization in the sense of Defini-tion 1.7, and the class of -special maps is dense in χ ∗ Iso .Proof. Lemma 7.7 implies that (7.30) is a cofibration and ψ × id in (7.29)is cocartesian over C > , with the same fibers as ψ in (7.24). Then ψ is alocalization by Lemma 2.10 (ii),(iv), and t is a localization since it has aright-adjoint η : C > → ar ‡ ( C > ) induced by (7.11). To see that 0-specialmaps are dense, note that the adjunction map Id → η ◦ t is cocartesian over111 r (∆ < ) o , so that the subclass v ⊂ t ∗ Iso of maps that are cocartesian over ar (∆ < ) o is dense in t ∗ Iso . Therefore ψ ∗ v is dense in χ ∗ Iso , and this is exactlythe class of 0-special maps. (cid:3) Now assume given a coadjoint pair of functors C ′ { adj } → C between2-categories C ′ , C , and let(7.31) γ : C ′ { eq } → C be the corresponding 0-special lax 2-functor of Lemma 7.4. We then havethe 2-categories γ ∗ C , γ ∗ C , γ ∗ C of (6.5), with identifications ι ∗ l ( γ ∗ C ) ∼ = γ ∗ l C , l = 0 , 1. We also have a 2-functor γ : C ′ → C and a lax 2-functor γ : C ′ → C ,with the decompositions (6.6) for γ , γ and γ . Proposition 7.9. The component e γ of the decomposition (6.6) of the lax -functor (7.31) factors as (7.32) C ′ { eq } e γ × id −−−−→ γ ∗ C{ eq } γ ⋄ −−−−→ γ ∗ C , where γ ⋄ is a -special lax -functor over eq / ∆ o equipped with an isomor-phism ι ∗ ( γ ⋄ ) ∼ = id . Moreover, such a factorization is unique up to a uniqueisomorphism.Proof. Denote W ( C ′ , γ ) = C ′ [0] { eq } ≀ γ [0] C , where the wreath product withrespect to a lax 2-functor is defined by (6.41), and γ [0] is the compositionof γ and the embedding C ′ [0] ⊂ C ′ . Let W ( C ′ , γ ) = W ( C ′ , γ ) × eq ≀ pt eq ≀ pt as in (7.28). Then we actually have W ( C ′ , γ ) ∼ = W ( γ ∗ C ), and on the otherhand, since γ is 0-special, the lax 2-functor W ( C ′ , γ ) → γ ∗ C of (6.42) is also0-special. By Remark 7.3, it then induces a functor(7.33) W ( γ ∗ C ) ∼ = W ( C ′ , γ ) ⊂ W ( C ′ , γ ) → γ ∗ C that sends 0-special maps to 0-special maps, and 0-anchor maps to 0-anchormaps. Therefore if we compose (7.33) with s : γ ∗ C → ( γ ∗ C ) > to obtain afunctor Φ( γ ) : W ( γ ∗ C ) → ( γ ∗ C ) > , then Φ( γ ) inverts 0-special maps, andtherefore by Corollary 7.8, uniquely factors through a functor(7.34) Θ( γ ) = χ ! Φ( γ ) : γ ∗ C → ( γ ∗ C ) > . If we let π l : ( γ ∗ l C ) > → ∆ o> , l = 0 , π ◦ Φ( γ ) ∼ = π ◦ χ , so again by uniqueness, Θ( γ ) is a functors over ∆ o> (in particular, it factors through γ ∗ C ⊂ ( γ ∗ C ) > ). Moreover, if we let a γ ∗ C , then maps in χ ∗ a cocartesian over eq are exactly 0-anchor maps, and Φ( γ ) sends those to mapscocartesian over ∆ o . Therefore Θ( γ ) is a lax 2-functor from γ ∗ C to γ ∗ C . Byvirtue of uniqueness, if γ factors through the projection C ′ { eq } → C ′ , sothat γ ∼ = γ , then Θ( γ ) ∼ = Id . Moreover, Φ( γ ) is obviously functorial withrespect to C ′ , and then so is Θ( γ ): for any 2-functor δ : C ′′ → C ′ , we have anatural isomorphism(7.35) Θ( γ ◦ ( δ × id )) ∼ = δ ∗ Θ( γ ) . Moreover, the functor W ( e γ ) : W ( C ′ ) → W ( γ ∗ C ) induced by e γ : C ′ → γ ∗ C fits into a commutative diagram W ( C ′ ) W ( e γ a ) −−−−→ W ( γ ∗ C ) y y Φ( γ ) C ′ { eq } s −−−−→ ( γ ∗ C ) > , where the left vertical arrow is again induced by (6.42), and again by Corol-lary 7.8, this induces an isomorphism(7.36) γ ∼ = Θ( γ ) ◦ e γ of lax 2-functors from C ′ to γ ∗ C .Now consider the map { , } × { , } → { , } sending l × l ′ to max( l, l ′ ), l, l ′ ∈ { , } , and let m : eq × eq → eq be the corresponding 2-functor.Then γ m = γ ◦ ( id × m ) defines a coadjoint pair of functors between C ′ { eq } = C ′ [ { , } ] and C , so that (7.34) provides a lax 2-functor(7.37) Θ( γ m ) : γ ∗ C{ eq } → γ ∗ C , where we identify γ ∗ m C ∼ = γ ∗ C{ eq } and γ ∗ m C ∼ = γ ∗ C . Moreover, say thata map f in W ( γ ∗ m C ) is 00 -special if it is cocartesian over eq , and χ ( f ) is0-special in γ ∗ m C = γ ∗ C{ eq } ; then Φ( γ m ) sends 00-special maps to 0-specialmaps, and therefore the lax 2-functor (7.37) is 0-special. Thus if we take γ ⋄ = Θ( γ ◦ ( id × m )), with the isomorphism γ ⋄ ◦ ( e γ × id ) ∼ = e γ provided by(7.36), we obtain the decomposition (7.32). Moreover, applying (7.35) tothe embedding ι : C ′ → C ′ { eq } , we obtain an isomorphism(7.38) ι ∗ ( γ ⋄ ) ∼ = Θ( γ m ◦ ( ι × id )) , and since ( id × m ) ◦ ( ι × id ) : C ′ { eq } → C ′ { eq } factors through the tautolog-ical projection C ′ { eq } → C ′ ∼ = ι ( C ′ ) ⊂ C{ eq } , the target of the isomorphism(7.38) is the identity functor. 113his proves existence. For uniqueness, assume given some other decom-position (7.32) of the functor e γ , and apply the construction above to theadjoint pair γ ⋄ . Then ( γ ⋄ ) = i ∗ γ ⋄ is identified with id , so that we have γ ⋄ ∼ = Θ( γ ⋄ ◦ ( id × m )) ◦ ( id × id ) = Θ( γ ⋄ ◦ ( id × m )), and then (7.35) for the2-functor δ = e γ × id provides an identification δ ∗ γ ⋄ ∼ = δ ∗ Θ( γ ⋄ ◦ ( id × m )) ∼ = Θ( γ ⋄ ◦ ( id × m ) ◦ ( δ × id )) = Θ( γ m ) . Since δ is dense, δ ∗ = id , so that this reads as γ ⋄ ∼ = Θ( γ m ). (cid:3) We note that in particular, the functor γ ⋄ in the decomposition (7.32)induces a lax 2-functor(7.39) Θ( γ ) = ι ∗ ( γ ⋄ ) : γ ∗ C → γ ∗ C , equipped with an isomorphism Θ( γ ) ◦ e γ ∼ = e γ , and this is the essentialingredient of the whole thing (we actually construct it first, in (7.34)). Wecall Θ( γ ) the twisting functor associated to the coadjoint pair. To computeits components Θ( γ ) [ n ] more explicitly, one can use (7.25) and (7.26). Thisshows that explicitly, the coadjoint pair γ defines an adjoint pair of maps h : γ ( c ) → γ ( c ), h ∨ : γ ( c ) → γ ( c ) for any c ∈ C ′ [0] , and for any [ m ] ∈ ∆,we have(7.40) Θ( γ ) [ m ] ( c ) = m Y l =1 h ∨ ( b ol ! c ) ◦ a ol ! ( c ) ◦ h ( b o ( l − c ) , c ∈ ( γ ∗ C ) [ m ] , where the product is the product (6.4), and b l : [0] → [ m ] is the embeddingonto l ∈ [ m ]. The maps (2.7) are induced by the adjunction maps between h and h ∨ , but writing them down explicitly is hard and probably pointless. As an application of Proposition 7.9, let usgive a somewhat more invariant description of the adjunction 2-category(7.2) of a 2-category C .As in Example 2.17, let Nat = tw (∆), with its natural cofibration s × t : Nat → ∆ o × ∆, and let Eq = ( V o × V ) ∗ tw (Sets), where V : ∆ → Sets isthe forgetful functor. Both Nat and Eq are ∆-kernels in the sense of Defini-tion 2.16, and V induces a morphism Nat → Eq . For any [ m ] ∈ ∆, the fiber Eq [ n ] of the cofibration t : Eq → ∆ is given by Eq [ n ] = ∆ o e ( V ([ n ])), and Nat [ n ] ⊂ Eq [ n ] is ∆ o [ n ] = (∆ / [ n ]) o , with the embedding induced by the tau-tological functor id : [ n ] → e ( V ([ n ])). In particular, Eq [1] = ∆ o e ( { , } ) = eq and Nat [1] = nat , with the embedding nat → eq given by the composition ofthe functors (7.4). 114hat we want to do is to construct a ∆-kernel Adj that fits in between Nat and Eq and completes a kernel version of (7.4). In order to do this,we generalize the description of the 2-category adj in terms of the diagrams(7.5) and (7.6) given in Subsection 7.1.To do this, consider the embedding ρ : ∆ + → ∆ with its left-adjoint λ : ∆ → ∆ + and the composition κ : ∆ → ∆, let b be the class of maps in∆ + that are bispecial, and define a cofibration C → ∆ by(7.41) C = λ ∗ Id b ∗∗ ρ ∗ Eq , where Eq is also considered as a cofibration over ∆. As in Definition 6.7,the fibers C [ n ] of the cofibration (7.41) can be described as in Example 2.14.By (3.10), we have [ n ] \ ∆ + ∼ = (∆ − / [ n ]) o , and since ∆ t ⊂ ∆ − , we thenhave an embedding v n : ∆ t / [ n ] → ∆ − / [ n ]. This embedding is fully faithfuland left-admissible, with the left-adjoint functor v † n sending a special arrow f : [ m ] → [ n ] to the anchor component a of its decomposition (1.8) for theanchor/bispecial fatorization system, and the adjunction map id → v n ◦ v † n is bispecial. Since ∆ t ∼ = N , we can identify [ n ] ∼ = (∆ t / [ n ]) o , and we thenhave(7.42) C [ n ] = Sec b ((∆ − /λ ([ n ])) o , p ∗ λ ([ n ]) ρ ∗ Eq ) ∼ = Sec( λ ([ n ]) , w ∗ n Eq ) , where we denote(7.43) w n = ρ ◦ p [ n ] ◦ v on : [ n ] → ∆ . Moreover, for any special map f : [ n ] → [ n ′ ], we have v † on ◦ f ∗ ∼ = f † ◦ v † on ′ ,where f ∗ is the functor (1.2), and by adjunction, this induces a map a ( f ) : w n ◦ f † = ρ ◦ p [ n ] ◦ v on ◦ f † → ρ ◦ p [ n ] ◦ f ∗ ◦ v on ′ ∼ = ρ ◦ p [ n ′ ] ◦ v on ′ = w n ′ . Then in terms of (7.42), the transition functor f ! : C [ n ] → C [ n ′ ] for a map f : [ n ] → [ n ′ ] is given by f ! = a ( λ ( f )) ! ◦ θ ( f ) ∗ , where θ is the functor (3.12),so that θ ( f ) = λ ( f ) † . Explicitly, w n : [ n ] → ∆ corresponds to the diagram(7.44) [ n ] t † −−−−→ [ n − t † −−−−→ . . . t † −−−−→ [1] t † −−−−→ [0]in ∆, with the maps t † : [ l ] → [ l − 1] adjoint to the embeddings t : [ l − → [ l ],and the fiber C [ n ] is opposite to the category of pairs h γ, π i of a functor γ : [ n +1] = λ ([ n ]) → ∆ and a map π : V ( γ ) → V ( w n +1 ). For any map f : [ n ] → [ n ′ ], the transition functor f ! : C [ n ] → C [ n ′ ] sends h γ, π i to θ ( f ) ∗ γ equipped with the composition map(7.45) V ( θ ( f ) ∗ γ ) V ( θ ( f ) ∗ π ) −−−−−−→ V ( θ ( f ) ∗ w n +1 ) V ( a ( λ ( f ))) −−−−−−−→ V ( w n ′ +1 ) . θ ( f ) is bispecial, both γ ( n + 1) and γ (0) with the projec-tion π (0) : V ( γ (0)) → V ( κ ([ n ])) are functorial with respect to h γ, π i . There-fore sending h γ, π i to γ (0) × γ ( n + 1) defines a functor χ : C → κ ∗ Eq × ∆ o cocartesian over ∆ o . We can then define a category Adj ′ by the cartesiansquare(7.46) Adj ′ −−−−→ C ν ′ y y χ Eq −−−−→ κ ∗ Eq × ∆ o , where the bottom arrow is the product of the full embedding a ! : Eq → κ ∗ Eq induced by the adjunction map a : id → κ , and the full embedding pt → ∆ o onto [0] ∈ ∆ o . Then ν ′ is obviously a cofibration, so that Adj ′ is a ∆-kernel and ν ′ is a morphism of ∆-kernels, while the full embedding Adj ′ ⊂ C is cocartesian over ∆. For any [ n ] ∈ ∆, the subcategory Adj ′ [ n ] ⊂ C [ n ] isspanned by pairs h γ, π i such that γ ( n + 1) = [0], and π factors through V ( w ′ n +1 ) ⊂ V ( w n +1 ), where w ′ n ⊂ w n is obtained by replacing [ n ] in (7.44)with t ([ n − ⊂ [ n ]. In particular, Adj ′ [0] is ∆ o , and adj ′ = Adj ′ [1] is the arrowcategory ar ( eq ).Now recall that by (7.5), adj is naturally identified with the full subcat-egory in ar ( eq ) spanned by bispecial 1-special maps. We can now denote by ε : pt → ∆ the embedding onto [1] ∈ ∆, so that we have adj ′ ∼ = ε ∗ Adj ′ , anddefine a category Adj by the cartesian square(7.47) Adj α −−−−→ Adj ′ y y ε ∗ adj −−−−→ ε ∗ adj ′ where the rightmost vertical arrow is the functor (2.22). Then by definition, Adj ⊂ Adj ′ is a full subcategory and a subcofibration over ∆, and its fiber Adj [ n ] ⊂ Adj ′ [ n ] is spanned by pairs h γ, π i such that f ! h γ, π i ∈ adj for any f : [ n ] → [1]. Since Adj [0] ∼ = Adj ′ [0] , the condition is empty if f factorsthrough [0], so that Adj [0] ∼ = ∆ o and Adj [1] ∼ = adj . The functor ν ′ of (7.46)induces a functor(7.48) ν : Adj → Eq , cocartesian over ∆, whose component ν [1] is identified with ν of (7.4). Lemma 7.10. The functor (7.48) is a cofibration, and α : Adj → Adj ′ of (7.47) admits a left-adjoint functor β : Adj ′ → Adj cocartesian over Eq . roof. For any [ n ] ∈ ∆, h γ, π i ∈ Adj ′ [ n ] , l ∈ [ n +1], m ∈ w ′ n +1 ( l ), denote γ ( l ) m = γ ( l ) \ π ( l ) − ( { , . . . , m − } ) ⊂ γ ( l ). This defines a decreasingfiltration γ ( l ) q on γ ( l ), and by the definition of the functor w ′ n +1 , the map g l : γ (0) → γ ( l ) fits into a cartesian square(7.49) γ (0) l g ′ l −−−−→ γ ( l ) y y γ (0) −−−−→ γ ( l ) , where the vertical arrows are the embeddings. If as in Subsection 7.1, weinterpret eq o as the category of injective arrows [ n ] → [ n ] in ∆, then (7.49)represents a object ar ( eq ) ∼ = adj ′ , and by (7.45), this is exactly the object f ! h γ, π i for the unique map f : [ n ] → [1] such that l = θ ( f )(1) ∈ [ n +1]. Then h γ, π i lies in Adj [ n ] iff for any l ∈ [ n +1], g l : γ (0) → γ ( l ) is bispecial and g ′ l isbijective, so that (7.49) is a square of the form (7.5).Now, as in the proof of Lemma 7.4, for any h γ, π i ∈ Adj [ n ] and l ∈ [ n +1],1 ≤ l ≤ n , we have a unique decomposition(7.50) γ (0) b l −−−−→ γ ′ ( l ) a l −−−−→ γ ( l )of the map g l such that b l is bispecial and strict and injective on γ (0) l in thesense of Remark 7.1, and a l is a composition of an anchor map and a mapthat is strict and injective on γ ′ ( l ) \ b l ( γ (0) l ). But then b l is also strict andinjective on γ (0) l +1 ⊂ γ (0) l , so that by the uniqueness of (7.50), the map γ ′ ( l ) → γ ( l ) → γ ( l + 1) factors through a l +1 : γ ′ ( l + 1) → γ ( l + 1). Then allthe maps b l : γ (0) → γ ′ ( l ) fit together into a functor γ ′ : [ n +1] → ∆ equippedwith a map a : γ ′ → γ , and h γ ′ , π ◦ V ( a ) i lies in Adj [ n ] ⊂ Adj ′ [ n ] . Thereforesending h γ, π i to h γ ′ , π ◦ V ( a ) i defines a functor β [ n ] : Adj ′ [ n ] → Adj [ n ] left-adjoint to α [ n ] : Adj [ n ] → Adj ′ [ n ] . By construction, γ (0) = γ ′ (0), so that β [ n ] is a functor over Eq [ n ] . Then by Lemma 2.11 (iii), Adj [ n ] → Eq [ n ] isa cofibration, and β [ n ] is cocartesian. Finally, again by construction, thefunctors β [ n ] commute with the transition functors f ! for all maps f in ∆,so that to finish the proof, it remains to apply Lemma 2.11 (ii). (cid:3) By Lemma 7.10, Adj is a ∆-kernel, and (7.48) is a morphism of ∆-kernels. In particular, Adj [ n ] → ∆ o is a cofibration for any [ n ] ∈ ∆. It turnsout that these cofibrations can be also constructed inductively, starting with Adj [1] = adj . Namely, assume given a cocartesian square (3.6) with l = 1,and denote p = s † : [ n ] → [1], q = t † : [ n ] → [ n − 1] (explicitly, p = id on s ([1]) ⊂ [ n ] and sends the rest to 1 ∈ [1], and q = id on t ([ n − ⊂ [ n ]117nd sends the rest to 0 ∈ [ n − τ : t ∗ w ′ n +1 → w n equal to id at l ∈ [ n ], l ≥ 1, and to q at l = 0, and sending h γ, π i to h t ∗ γ, τ ◦ t ∗ π i defines a functor t ∗ : Adj [ n ] → Adj [ n − . Lemma 7.11. For any n ≥ , we have a cartesian square (7.51) Adj [ n ] t ∗ × p ! −−−−→ Adj [ n − { adj } ν [ n ] y y s ◦ ( ν [ n − × id ) Eq [ n ] q ! × p ! −−−−→ Eq [ n − × eq , where s is the projection (7.12) .Proof. The cocartesian square (3.6) induces a cocartesian square[0] > s > −−−−→ [ l ] >t > y y t > [ n − l ] > s > −−−−→ [ n ] > , and a functor γ : [ n +1] = [ n ] > → I to any category I is uniquely definedby its restrictions γ ′ = t > ∗ γ : [ l ] > → I , γ ′′ = s > ∗ γ : [ n − l ] > → I , and anisomorphism t > ∗ γ ′′ ∼ = s > ∗ γ ′ . In particular, we can take I = ∆. Moreover, s > and t > are antispecial, thus have right-adjoint functors s > † , t > † , and givingtwo maps π ′ : V ( γ ′ ) → V ( w ′ l +1 ), π ′′ : V ( γ ′′ ) → V ( w ′ n − l +1 ) is equivalent togiving a single map(7.52) π = π ′ × π ′′ : V ( γ ) → ( t > † ) ∗ V ( w ′ l +1 ) × ( s > † ) ∗ V ( w ′ n − l +1 ) . For any set S , denote by S > : [0] > → Sets the functor sending 0 ∈ [0] > to S and o ∈ [0] > to pt . Then we have s > ∗ V ( w ′ l +1 ) ∼ = V ([ l ]) > , t > ∗ V ( w n − l +1 ) ∼ = V ([ n − l ]) > , and if we let a = s ◦ t = t ◦ s : [0] → [ n ] be the embedding onto l ∈ [ n ], and then take l = 1, we have a cartesian square V ( w ′ n +1 ) −−−−→ ( t > † ) ∗ V ( w ′ ) × ( s > † ) ∗ V ( w ′ n ) y y ( a > † ) ∗ V ([ n ]) > −−−−→ ( a > † ) ∗ V ([1]) > × ( a > † ) ∗ V ([ n − > , where the bottom arrow is induced by p × q : V ([ n ]) → V ([1]) × V ([ n − Adj ′ , adj ′ Adj , adj , and then the explicit description of Adj ⊂ Adj ′ in termsof diagrams (7.49) shows that it induces (7.51) on the nose. (cid:3) Lemma 7.11 immediately implies by induction that for any [ n ], the cofi-bration Adj [ n ] → ∆ o is a 2-category, and ν [ n ] : Adj [ n ] → Eq [ n ] is a 2-functor.The functor (7.48) has an obvious fully-faithful right-adjoint(7.53) η : Eq → Adj . Explicitly, an object in Eq [ n ] ⊂ Eq is a pair h [ m ] , π i , [ m ] ∈ ∆, π : V ([ m ]) → V ([ n ]), and η ( h [ m ] , π i ) is obtained by taking the constant functor γ : [ n ] → ∆ with value [ m ], extending it to a functor γ > : [ n +1] = [ n ] > → ∆ sending o ∈ [ n ] > to [0], and then extending π to a map π : V ( γ > ) → V ( w n +1 ).While η is not cocartesian over the whole ∆, it is obviously cocartesian overall antispecial maps. Over [1] ∈ ∆, η [1] : eq → adj is the lax 2-functor (7.11),and for any [ n ], we have the base change isomorphism(7.54) e ◦ η [ n ] ∼ = ( η [ n − × id ) ◦ η ◦ e, induced by (7.51), so that by induction, η [ n ] is a lax 2-functor for any n .In general, the projection ν does not have a left-adjoint. However, if wedefine a ∆-kernel Adj by the cartesian square Adj −−−−→ Adj ν y y ν Nat −−−−→ Eq , then ν does have a left-adjoint δ : Nat → Adj . Namely, Adj ⊂ Adj isspanned by pairs h γ, π i such that π = V ( π ′ ) for a map π ′ : γ → w ′ n +1 , sothat for any [ n ] ∈ ∆, we have an object h w ′ n +1 , V ( id ) i ∈ Adj n ] . Then objectsin Nat are arrows f : [ m ] → [ n ] in ∆, and δ sends f to f ! h w ′ n +1 , V ( id ) i .By construction, δ is cocartesian over ∆, and it is easy to check usingLemma 7.10 that it is also cocartesian over ∆ o . Therefore composing δ with the embedding Adj → Adj , we obtain a functor δ : Nat → Adj that fitsinto a diagram(7.55) Nat δ −−−−→ Adj ν −−−−→ Eq of morphisms of ∆-kernels. Over [1] ∈ ∆, it reduces to (7.4). Now for any2-category C , let(7.56) A dj ( C ) = ( Adj ⊗ ∆ C ) ♮ ⊂ Adj ⊗ ∆ C o with fibers Fun ( Adj [ m ] , C ) and transition func-tors induced by those of the cofibration Adj → ∆. By Example 2.17, themorphism δ of (7.55) induces a functor A dj ( C ) → C ♮ . Proposition 7.12. The cofibration A dj ( C ) → ∆ o is a -category, and thefunctor A dj ( C ) → C ♮ induced by the morphism δ of (7.55) identifies it withthe -category (7.2) .Proof. The second claim is by definition true over [0] and [1] ∈ ∆, so itfollows from the first. Also by definition, A dj ( C ) [0] = C [0] is discrete, sowhat we need to check is the Segal condition. By induction on n , it sufficesto consider the squares (3.6) with l = 1. Consider the 2-category Adj [ n ] ,with the embeddings s : adj = Adj [1] → Adj [ n ] , t : Adj [ n − → Adj [ n ] , and the embedding e : Adj [ n ] → Adj [ n − { adj } of (7.51). Denote by S = V ([ n − Adj [ n − , with the embedding S = Adj [ n − , [0] → Adj [ n − , and let p : S × adj → Adj [ n ] be the adjointpair given by s on 0 × adj and by the constant projection onto i on i × adj , i ∈ S \ { } . Then by Proposition 7.9 and Lemma 7.4, p extends uniquelyto a 2-functor p : Adj [ n − { adj } → Adj [ n ] such that p ◦ e ∼ = id . Now assumegiven a 2-functor F : Adj [ n − → C and an adjoint pair γ : adj → C with γ (1) = F (0), and let E be the groupoid of 2-functors F ′ : Adj [ n ] → C equipped with isomorphisms s ∗ F ′ ∼ = γ and t ∗ F ′ ∼ = F . Extend γ to an adjointpair γ ′ : S × adj → C given by γ on 0 × adj and by the constant projectiononto F ( i ) ∈ C [0] on i × adj , i ∈ S \ { } . Let l : Adj [ n − → Adj [ n − { adj } be induced by the embedding pt → adj onto 1 ∈ adj [0] , let r : S × adj = S { adj } → Adj [ n − { adj } be induced by the embedding S → Adj [ n − , andlet P be the category of 2-functors F ′′ : Adj [ n − { adj } → C equipped withisomorphisms l ∗ F ′′ ∼ = F , r ∗ F ′′ ∼ = γ ′ . Then the 2-functors e and p inducefunctors p ∗ : E → P , e ∗ : P → E such that e ∗ ◦ p ∗ ∼ = Id , so that E is aretract of P . But by Lemma 7.4 and Proposition 7.9, we have P ∼ = pt . (cid:3) Fix a category C equipped with a unitalmonoidal structure B C in the sense of Definition 6.18. Recall that for any set S , we have the small category e ( S ) of Subsection 3.1 (objects are elements s ∈ S , and there is exactly one morphism between any two objects).120 efinition 8.1. A a small C -enriched category with a set of objects S isa lax 2-functor A from e ( S ) to B C . A functor between small C -enrichedcategories h S, A i and h S ′ , A ′ i is a pair h f, g i of a map f : S → S ′ and amorphism g : A → ∆ o ( e ( f )) ∗ A ′ . Remark 8.2. Explicitly, defining A amounts to giving an object A ( s, s ′ ) in B C [1] ∼ = C for any s, s ′ ∈ S and equipping them with the composition andunity maps; thus Definition 8.1 is a repackaging of the usual notion of acategory enriched over C . Example 8.3. If S = pt is a one-element set, then a C -enriched category h S, A i is the same thing as a unital associative algebra object in C . Inparticular, the unit object 1 ∈ C is tautologically an algebra, and it definesthe point C -enriched category pt C . For any h S, A i and object s ∈ S , we havethe tautological functor i s : pt C → h S, A i given by the embedding pt → S onto s and the unit map 1 → A ( s, s ). Definition 8.4. The opposite category h S, A i o to a small C -enriched cate-gory h S, A i is the C ι -enriched category h S, A i o = h S, A ι i , where A ι = ι ◦ A ,and C ι is C with the opposite monoidal structure B C ι = ι ∗ B C .Assume given a small C -enriched category h S, A i . To define modules over A , recall that we have the functor κ : ∆ → ∆ equipped with the adjunctionmap a : id → κ of (3.8). The transition functor a o ! for the opposite map thenprovides a functor a o ! : κ o ∗ B C → C cocartesian over ∆ o . Definition 8.5. A right module M over a small C -enriched category h S, A i is a functor M : ∆ o e ( S ) → κ o ∗ B C over ∆ cocartesian over all left-anchormaps s o and equipped with an isomorphism a o ! ◦ M ∼ = A . Remark 8.6. More generally, one can replace κ o ∗ B C with a C -module M in the sense of Definition 6.30, and develop the theory of h S, A i -moduleswith values in M . We will not need this.Recall that ∆ o e ( S ) [0] is the set S itself, so that a right A -module M defines an object M ( s ) ∈ B C [1] ∼ = C for any s ∈ S ; the rest of the structureequips these objects with the right actions of A ( s, s ′ ) ∈ C . The correspon-dence M M ( s ) is functorial in M , in that we have a functor(8.1) A -mod → C , M M ( s ) , where A -mod is the category of right A -modules, and a similar functor forleft A -modules. 121 emma 8.7. For any small C -enriched category h S, A i , the functor (8.1) admits a left-adjoint functor C → A -mod , V V s .Proof. For any small category I and object i ∈ I , let ∆ o + I i = (∆ o I ) i =∆ o + N ( I ) i , where (∆ o I ) i resp. N ( I ) i are as in (6.13) resp. (3.17), and recallthat if i ∈ I is initial, then λ o ∗ ∆ o + I i ∼ = ∆ o I . As in Lemma 3.2 and in (6.14),this gives an embedding λ ( i ) : ∆ o I → ∆ o + I left-adjoint to the embedding ρ ( i ) : ∆ o + I i ⊂ ρ o ∗ ∆ o I → ∆ o I , and the opposite functor λ o ( i ) is then right-adjoint to the opposite functor ρ o ( i ).Now recall that every s ∈ e ( S ) is initial, and denote by A -mod s thecategory of functors M : ∆ o + e ( S ) s → ρ o ∗ κ o ∗ B C over ∆ o + , cocartesian overall anchor maps and equipped an isomorphism ρ o ∗ ( a o ! ◦ M ) ∼ = ρ o ∗ A . Then theadjunction map ρ ◦ κ = ρ ◦ λ ◦ ρ → ρ provides an identification ρ o ∗ κ o ∗ B C ∼ = C × ρ o ∗ B C , with the projection onto the second factor given by a o ! , so that M ∈ A -mod s is completely defined by its first component(8.2) M s : ∆ o + e ( S ) s → C that has to be cocartesian over all anchor maps. Since ∆ o + e ( S ) s has a ter-minal object o , and the map e → o is an anchor map for any e ∈ ∆ o + e ( S ) s ,evaluating at o gives an equivalence A -mod s ∼ = C . In terms of this equiva-lence, the functor (8.1) is given by the pullback functor ρ o ( s ) ∗ . Since ρ o ( s )has a right-adjoint λ o ( s ), the left Kan extension ρ o ( s ) ∆ o ! exists and is givenby (2.6) that reads as(8.3) ρ o ( s ) ∆ o ! ∼ = a o ! λ o ( s ) ∗ . Since a is an anchor map and A is a lax 2-functor, this implies that(8.4) ρ o ( s ) ∆ o ! ρ o ( s ) ∗ A ∼ = A, and then for any V ∈ C that corresponds to some V + s under the equivalence C ∼ = A -mod s , the functor(8.5) V s = ρ o ( s ) ∆ o ! V + s = a o ! λ o ( s ) ∗ V + s comes equipped with an isomorphism a o ! ◦ V s ∼ = ρ o ( s ) ∆ o ! ( a o ! ◦ V + s ) ∼ = ρ o ( s ) ∆ o ! ρ o ( s ) ∗ A ∼ = A, thus defines an h S, A i -module. Sending V to V s ∈ A -mod then gives our leftadjoint. (cid:3) A -modules, consider the nerve N ( e ( S )) of the small category e ( S ), and note that the left Kan extension κ o ! N ( e ( S )) exists and is naturally identified with the nerve N ( e ( S ) < ). Thecategory e ( S ) < is in turn naturally identified with e ( S + / [1]), where S + = S ⊔{ o } is equipped with the map S + → [1] sending o to 0 ∈ [1] and S ⊂ S + to1 ∈ [1], so we have a projection τ : e ( S ) < = e ( S + / [1]) → [1]. This projectioninduces a projection ∆ o e ( S + / [1]) → ∆ o [1] = nat ⊂ eq , so it makes sense tospeak about 1-special maps in ∆ o e ( S + / [1]) in the sense of Subsection 7.2.Then by adjunction, right A -modules M correspond bijectively to 1-speciallax 2-functors(8.6) M † : ∆ o e ( S + / [1]) → B C equipped with an isomorphism M † | ∆ o e ( S ) ∼ = A . In these terms, an element s ∈ S with the embedding map i s : s → S defines a section i For any enriched category h S, A i over C , a right A -module M is representable if M ∼ = 1 s for some s ∈ S , where 1 ∈ C is the unit object,and polyrepresentable if M ∼ = V s , s ∈ S , V ∈ C , where V s and 1 s are as in(8.5). Moreover, M is ind-representable resp. ind-polyrepresentabe if it is afiltered colimit of representable resp. polyrepresentable modules. Example 8.9. Assume given a commutative ring k , and let C = k -modbe the category of k -modules, with its usual tensor structure. Then a C -enriched small category A is the same thing as a k -linear small categoryin the sense of Subsection 1.4, and right A -modules are k -linear functors A → k -mod. Representable modules correspond to representable functors.If A is a k -algebra considered as a k -linear category with one object, thenright modules are the right modules in the usual sense, and a right module ispolyrepresentable resp. ind-polyrepresentable iff it is free (that is, of the form V ⊗ k A for some V ∈ k -mod) resp. flat. If we consider the category P ( A )of finitely generated projective left modules over a flat k -algebra A , then P ( A )-modules are the same thing as A -modules, and a right P ( A )-moduleis representable resp. ind-representable iff the corresponding A -module M is finitely generated projective resp. flat.We will also need a covariant version of the Yoneda Lemma similar to(1.26) and (1.27). Namely, for any small C -enriched category h S, A i , define a left module N over h S, A i as a right module over the opposite category h S, A i ι N is given by a functor N : ∆ o e ( S ) → κ o ∗ ι B C over ∆ o cocartesian over all right-anchor maps and equipped withan isomorphism a oι ! ◦ N ∼ = A . Now, the functorial square (3.9) induces anequivalence κ o ∗ ♭ B C ∼ = κ o ∗ B C × B C κ ∗ oι B C , and for any right h S, A i -module M ,we can consider the functor(8.7) M ⊠ A N = ζ ( o ) ◦ ( M × A N ) : ∆ o e ( S ) → κ o ∗ ♭ B C o ∼ = B C κ o♭ ( o ) ∼ = C , where we extend κ ♭ to ∆ < , and ζ ( o ) is the functor (2.1) for the cofibra-tion κ o ∗ ♭ B C → ∆ o> . We then define the tensor product M ⊗ A N ∈ C ascolim ∆ o e ( S ) M ⊠ A N , if the colimit exists. Lemma 8.10. For any C -entiched category h S, A i , left h S, A i -module N ,and ind-polyrepresentable right h S, A i -module M , the tensor product M ⊗ A N exists, and if M ∼ = V s is polyrepresentable, then M ⊗ A N ∼ = V ⊗ N ( s ) .Proof. Since colimits over ∆ o e ( S ) commute with filtered colimits, it sufficesto consider the case when M = V s is polyrepresentable. By (8.5), we have M ∼ = ρ o ( s ) ∆ o ! V + s , where the Kan extension functor ρ o ( s ) ∆ o ! is given by (8.3).Since the map a is right-anchor, and both A and N are cocartesian overright-anchor maps, we have A ∼ = ρ o ( s ) ∆ o ! ρ o ( s ) ∗ A , N ∼ = ρ o ( s ) ∆ o ! ρ o ( s ) ∗ N , and M ⊠ A N ∼ = ρ o ( s ) ∆ o ! ( ζ ( o ) ◦ ( V + s × ρ o ( s ) ∗ A ρ ( s ) ∗ N )) , so that M ⊗ A N ∼ = colim ∆ o + e ( S ) s ( ζ ( o ) ◦ ( V + s × ρ o ( s ) ∗ A ρ ( s ) ∗ N )) , Since ∆ o + e ( S ) s has a terminal object h [0] , s i , the colimit exists and coincideswith N ( s ). (cid:3) -categories: the construction. Non-empty small C -en-riched categories h S, A i form a category that we denote by Cat( C ). In thesituation of Example 8.9, more is true: small k -linear categories and k -linear functors between them form a 2-category, and one can define a larger“Morita 2-category” by allowing general bimodules instead of functors. Toconstruct these 2-categories formally and in full generality, it is convenientto encode the fibers of the corresponding cofibrations over ∆ o by using theGrothendieck construction (that is, iterating the cylinder construction ofExample 2.1). This uses augmented sets of Subsection 3.1.124 efinition 8.11. For any monoidal category C and any pre-ordered set J ,a J -augmented C -enriched small category is the pair h S/J, A i of a proper J -augmented set S and a lax 2-functor A : ∆ o e ( S/J ) → B C . Example 8.12. For any order-preserving map f : J ′ → J and small C -enriched category h S/J, A i , we have the induced J ′ -augmented set f ∗ S = S × fJ J ′ , with the natural map f S : f ∗ S → S , and the induced small C -enriched category f ∗ h S/J, A i = h f ∗ S/J ′ , ∆ o ( e ( f S )) ∗ A i .We will be mostly interested in [ n ]-augmented enriched categories forordinals [ n ] ∈ ∆. To consider them all at once, denote by Sets ′ ⊂ Sets thefull subcategory of non-empty sets, and consider the cofibrations E Sets ′ ⊂ E Sets → ∆ o of Example 6.4. For any [ n ] ∈ ∆, the fiber E Sets [ n ] isnaturally identified with the category Sets / [ n ] of [ n ]-augmented sets, and E Sets ′ [ n ] ⊂ E Sets [ n ] is spanned by proper [ n ]-augmented sets. We alsohave the cofibration ϕ : E Sets + → E Sets ′ of (6.8), and for any proper[ n ]-augmented set S/ [ n ], the right comma-fiber S \ ϕ E Sets + is naturallyidentified with the simplicial replacement ∆ o e ( S/ [ n ]) of the correspondingcategory e ( S/ [ n ]). We can then consider the category e ∗ ϕ a ∗∗ π ∗ B C , where π : E Sets + → ∆ o is the structural cofibration, a is the class of cocartesianliftings of anchor maps, and e : E Sets ′ Id → E Sets ′ is induced by the em-bedding Sets ′ Id → Sets ′ . By definition, e ∗ ϕ a ∗∗ π ∗ B C is cofibered over E Sets ′ Id ,hence also over ∆ o , so we can replace it with its tightening, and let Aug( C ) bethe reduction of the resulting cofibered category in the sense of Remark 6.5.For any [ n ] ∈ ∆, the objects in the fiber Aug( C ) [ n ] are [ n ]-augmented C -enriched small categories in the sense of Definition 8.11. The transition func-tor f o ! corresponding to a map f : [ n ′ ] → [ n ] in ∆ sends an [ n ]-augmented C -enriched category h S/ [ n ] , A i to f ∗ h S/ [ n ] , A i = h f ∗ S/ [ n ′ ] , ∆ o ( f S ) ∗ A i , where f ∗ S = S × f [ n ] [ n ′ ], with the natural map f S : e ( f ∗ S/ [ n ′ ]) = f ∗ e ( S/ [ n ]) → e ( S/ [ n ]). The evaluation functor (2.23) induces a functor(8.8) ev : e ∗ E Sets + × E Sets ′ Id Aug( C ) → B C over ∆ o , and it is cocartesian over all anchor maps. Definition 8.13. For any unital monoidal category C , a C -enrichment ofa 2-category C ′ is the pair h S, A i of a functor S : C ′ [0] → Sets ′ and a lax2-functor A : C ′ [ S ] → B C , where C ′ [ S ] is the 2-category (6.7). Example 8.14. A C -enrichment of the trivial 2-category pt is the samething as a small C -enriched category h S, A i in the sense of Definition 8.1.125 emma 8.15. For any unital monoidal category C and -category C ′ equip-ped with a C -enrichment h S, A i in the sense of Definition 8.13, there existsa unique pair of a functor (8.9) Y ( S, A ) : C ′ → Aug( C ) , cocartesian over E Sets Id , and an isomorphism Y ( S, A ) ∗ ev ∼ = A , where ev isthe functor (8.8) .Proof. Immediately follows from (2.24). (cid:3) Now note that for any partially ordered set J and J -augmented C -enriched category h S/J, A i in the sense of Definition 8.11, Definition 8.5 canbe repeated verbatim with e ( S ) replaced by e ( S/J ), so that in particular,we have the notion of a right h S/J, A i -module M . For any order-preservingmap f : J ′ → J and right h S/J, A i -module M , f ∗ M = ∆ o ( f S ) ∗ M is a rightmodule over f ∗ h S/J ′ , A i . Moreover, if J has the smallest element o ∈ J ,then any s ∈ S o ⊂ S in the fiber over o ∈ J is still an initial object in e ( S/J ), so that Lemma 8.7 works with the same proof, and we have theobjects V s , V ∈ C , s ∈ S o ⊂ S . We can then repeat Definition 8.8 verbatim(but limited to s ∈ S o ⊂ S ). Now, for any j ∈ J ′ , we have the full embed-ding ε j : j \ J → J of the right comma-fiber j \ J , and for any j ′ ≥ j , wehave an embedding ε j,j ′ : j ′ \ J → j \ J . Then any j ∈ J is an initial objectin j \ J , and we can make the following. Definition 8.16. For any partially ordered set J , an J -augmented small C -enriched category h S/J, A i is a iterated cylinder resp. polycylinder resp. ind-cylinder resp. ind-polycylinder if for any j ≤ j ′ ∈ J and representable right ε ∗ j h S/J, A i -module M , the right ε ∗ j ′ h S/J, A i -module ε ∗ j,j ′ M is representableresp. polyrepresentable resp. ind-representable resp. ind-polyrepresentable. Remark 8.17. Explicitly, Definition 8.16 means that if h S/J, A i is an iter-ated polycylinder, then for any j ≤ j ′ ∈ J and s ∈ S j , we have ε ∗ j,j ′ s ∼ = V s ′ for some V ∈ C and s ′ ∈ S j ′ . But then for any V ′ ∈ C , we have ε ∗ j,j ′ V ′ s ∼ =( V ′ ⊗ V ) s , so that a polyrepresentable module also restricts to a polyrep-resentable module. Moreover, ε ∗ j,j ′ commutes with filtered colimits, so thatthe same holds for ind-cylinders and ind-representable modules, and thenind-polycylinders and ind-polyrepresetable modules. Example 8.18. If C = Sets is the category of sets, with the cartesian prod-uct, then a J -augmented small C -enriched category is the same thing as a126mall category A equipped with a functor A → J with non-empty fibers,and it is an iterated cylinder if and only if the functor is a cofibration. For J = [ n ], this is the same notion as in Subsection 3.4. Example 8.19. For any small C -enriched category h S, A i and right h S, A i -module M , the corresponding lax 2-functor (8.6) defines a small [1]-aug-mented C -enriched category. It is a cylinder resp. a polycylinder if M isrepresentable resp. polyrepresentable. We also have a small [1]-augmented C -enriched category defined by ι ∗ M † : ∆ o e ( S + / [1]) o → B C . Since all modulesover the unit enriched category pt C of Example 8.3 are polyrepresentable, ι ∗ M † is always a polycylinder; however, it is cylinder only if M ( s ) ∼ = 1 ∈ C for any s ∈ S . Proposition 8.20. Let C at ( C ) ⊂ Aug( C ) resp. M or ( C ) ⊂ Aug( C ) be thefull subcategories spanned by cylinders resp. polycylinders. Then the in-duced projections C at ( C ) , M or ( C ) → ∆ o are -categories in the sense ofDefinition 6.1. Moreover, assume that C has all filtered colimits, and thetensor product preserves them. Then the same holds for the full subcate-gories C at ∗ ( C ) and M or ∗ ( C ) in Aug( C ) spanned by ind-cylinders and ind-polycylinders. The proof of Proposition 8.20 is somewhat technical, so we postpone ituntil Subsection 8.3, and first make two remarks that do not depend on thespecifics of the proof. Firstly, a lax monoidal functor γ : C ′ → C betweenunital monoidal categories C , C ′ obviously induces a functor Aug( C ′ ) → Aug( C ) cocartesian over ∆ o . Moreover, γ sends cylinders to cylinders, andind-cylinders to ind-cylinders if it is continuous, so that we have 2-functors(8.10) γ : C at ( C ′ ) → C at ( C ) , γ : C at ∗ ( C ′ ) → C at ∗ ( C ) . If γ is not just lax monoidal but monoidal, then it also sends polycylinders topolycylinders, and ind-polycylinders to ind-polycylinders if it is continuous,so that we have 2-functors(8.11) γ : M or ( C ′ ) → M or ( C ) , γ : M or ∗ ( C ′ ) → M or ∗ ( C ) . Secondly, for a fixed C and any [ n ] ∈ ∆, C -enrichments of e ([ n ] / [ n ]) ∼ = [ n ] arelax 2-functors from [ n ] to B C . In particular, a 2-functor from [ n ] to B C givesan enrichment, and these correspond to objects in B C [ n ] . This gives a functor B C → Aug( C ) cocartesian over ∆ o . Any such [ n ]-augmented C -enriched127ategory is tautologically an iterated polycylinder, so the embedding factorsthrough M or ( C ) ⊂ Aug( C ), and we get a fully faithful 2-functor(8.12) pt C : B C → M or ( C ) ⊂ M or ∗ ( C ) . Informally, it identifies B C with the full 2-subcategory in M or ( C ) spannedby the unit enriched category pt C of Example 8.3. -categories: the proof. We now prove Proposition 8.20.For any [ n ] ∈ ∆, [ n ]-augmented C -enriched category h S/ [ n ] , A i , and l ∈ [ n ],we have the C -enriched category h S l , A l i = a ∗ l h S/ [ n ] , A i , where a l : [0] → [ n ]is the embedding onto l . Assume given such a h S/ [ n ] , A i , and consider theembedding t : [ n − → [ n ]. Lemma 8.21. The following conditions are equivalent. (i) h S/ [ n ] , A i is an iterated cylinder, resp. polycylinder, resp. ind-polycy-linder, resp. ind-polycylinder. (ii) t ∗ h S/ [ n ] , A i is an iterated cylinder, resp. polycylinder, resp. ind-poly-cylinder, resp. ind-polycylinder, and for any representable h S/ [ n ] , A i -module s , s ∈ S , the module M s = t ∗ s is representable, resp. poly-representable, resp. ind-representable, resp. ind-polyrepresentable.Proof. For any order-preserving map f : J ′ → J , the pullback f ∗ of Ex-ample 8.12 obviously preserves the cylinder conditions of Definition 8.16, so(i) ⇒ (ii) is obvious. In the other direction, we have to check a condition forany j ≤ j ′ ∈ [ n ]. If j ≥ 1, it follows from the first part of (ii), if j = 0 and j ′ = 1, it is the second part on the nose, and if j = 0 but j ′ ≥ 2, combineboth parts and Remark 8.17. (cid:3) To see explicitly the modules M s of Lemma 8.21 (ii), it is convenient touse the cofibration (3.19) associated to the left-closed embedding e ( S ) ⊂ e ( S/ [ n ]). Its fibers are given by (3.21), and in particular, for any s ∈ S , thefiber over h [0] , s i is naturally identified with ∆ o> e ( t ∗ S/ [ n − j s : ∆ o e ( t ∗ S/ [ n − → ∆ o e ( S/ [ n ]) that fits into a commutativesquare(8.13) ∆ o e ( t ∗ e ( S/ [ n − j s −−−−→ ∆ o e ( S/ [ n ]) y y ∆ o κ o −−−−→ ∆ o , M s ∼ = j ∗ s A, where j ∗ s A : ∆ o e ( t ∗ S/ [ n − → κ o ∗ B C is induced by the lax 2-functor A .Now consider a square (3.6) and the category ∆ o [ n ] ∼ = (∆ / [ n ]) o , and let∆ o [ n ] l ⊂ ∆ o [ n ] be the full subcategory spanned by maps f : [ m ] → [ n ] whoseimage contains l ∈ [ n ]. Moreover, (3.6) induces left-closed full embeddings∆ o [ l ] → ∆ o [ n ], ∆ o [ n − l ] → ∆ o [ n ]; let ∆ o [ n ] ≤ l , ∆ o [ n ] ≥ l ⊂ ∆ o [ n ] be theiressential images, and let(8.15) ∆ o [ n ] l = ∆ o [ n ] ≤ l ∪ ∆ o [ n ] ≥ l , ∆ o [ n ] l = ∆ o [ n ] l ∪ ∆ o [ n ] l . For any [ n ]-augmented set S/ [ n ], the simplicial replacement ∆ o e ( S/ [ n ])comes equipped with a projection π : ∆ o e ( S/ [ n ]) → ∆ o [ n ], and we canlet ∆ o e ( S/ [ n ]) l = ∆ o [ n ] l × ∆ o [ n ] ∆ o e ( S/ [ n ]), and similarly for ∆ o e ( S/ [ n ]) ≤ l ,∆ o e ( S/ [ n ]) ≥ l , ∆ o e ( S/ [ n ]) l , and ∆ o e ( S/ [ n ]) l . If we denote S ≤ l = s ∗ S , S ≥ l = t ∗ S , then we have ∆ o e ( S/ [ n ]) ≤ l ∼ = ∆ o e ( S ≤ l / [ l ]), ∆ o e ( S/ [ n ]) ≥ l ∼ =∆ o e ( S ≥ l / [ n − l ]), and(8.16) ∆ o e ( S l ) ∼ = ∆ o e ( S/ [ n ]) ≤ l ∩ ∆ o e ( S/ [ n ]) ≥ l ⊂ ∆ o e ( S/ [ n ]) l . We also have full embeddings(8.17) ∆ o e ( S/ [ n ]) l α −−−−→ ∆ o e ( S/ [ n ]) l β −−−−→ ∆ o e ( S/ [ n ])over ∆ o . The projection π : ∆ o e ( S/ [ n ]) l → ∆ o is a cofibration but not a2-category. The projection π : ∆ o e ( S/ [ n ]) l → ∆ o is not even a cofibration,but it is a cofibration over all anchor maps, and the embeddings (8.17) arecocartesian over anchor maps. By abuse of terminology, we will use theterm lax -functor to also signify a functor from ∆ o e ( S/ [ n ]) l or ∆ o e ( S/ [ n ]) l over ∆ o that is cocartesian over anchor maps. Note that by Example 3.3,giving such a lax 2-functor A : ∆ o e ( S/ [ n ]) l → B C is equivalent to givinglax 2-functors A ≤ l : ∆ o e ( S/ [ n ]) ≤ l → B C , A ≥ l : ∆ o e ( S/ [ n ]) ≥ l → B C and anisomorphism between their restrictions to the subcategory (8.16). Lemma 8.22. (i) In the notation and assumptions above, for any lax -functor A : ∆ o e ( S/ [ n ]) l → B C , the right Kan extension α ∗ A existsand is a lax -functor, while for any lax -functor A : ∆ o e ( S/ [ n ]) l → B C , the adjunction map A → α ∗ α ∗ A is an isomorphism. Assume that h s ∗ S/ [ l ] , A ≤ l i is an iterated ind-polycylinder, and let A = α ∗ A . Then the left Kan extension β ∆ o ! A exists and is a lax -functor. (iii) Finally, for any iterated ind-polyclynider h S/ [ n ] , A i , the adjunctionmap β ∆ o ! β ∗ A → A is an isomorphismProof. For (i), note that for any [ m ] ∈ ∆ and anchor map f : [ m ] → [ n ],the embedding ∆ o> e ( f ∗ S/ [ m ]) → ∆ o> e ( S/ [ n ]) admits a left-adjoint functor µ ( f ) provided by Lemma 3.8. In particular, this applies to the maps s , t , a = s ◦ t = t ◦ s in (3.6), and if we let µ ≤ l = µ ( s ), µ ≥ l = µ ( t ), µ l = µ ( a ),then any object x ∈ ∆ o> e ( S/ [ n ]) fits into a commutative square(8.18) x −−−−→ µ ′′ ( x ) y y µ ′ ( x ) −−−−→ µ l ( x ) . If x lies in ∆ o e ( S/ [ n ]) l \ ∆ o e ( S/ [ n ]) , then (8.18) is actually a square in∆ o e ( S/ [ n ]) l ⊂ ∆ o> e ( S/ [ n ]), and all its maps are anchor maps. Moreover,the oppposite square defines a standard functor c : V → ∆ e ( S/ [ n ]) l in thesense of Remark 6.3 equipped with an exact x -augmentation. The corre-sponding embedding c : V → ∆ e ( S/ [ n ]) l /x of (1.11) is left-admissible, so weobtain a framing for the functor α o in the sense of Lemma 1.14, and if weuse this framing to compute A = α ∗ A , we see that it exists by Remark 6.3.Moreover, the square (8.18) provides an isomorphism(8.19) A ( x ) ∼ = A ≤ l ( µ ≤ l ( x )) × A l ( µ l ( x )) A ≥ l ( µ ≥ l ( x )) , where A l : ∆ o e ( S l ) → B C is the common restriction of A ≤ l and A ≥ l . Thesquare (8.18) is functorial in x , and since µ l , µ ≤ l , µ ≥ l send anchor maps toanchor maps, A is cocartesian over anchor maps. Again by Remark 6.3, itis also unique with this property.For (ii), let L resp. R be the classes of those maps in ∆ o [ n ] ∼ = (∆ / [ n ]) o that are bijective over l ∈ [ n ] resp. [ n ] \ { l } ⊂ [ n ], and note that byExample 1.2, h L, R i is a factorization system on ∆ o [ n ]. Moreover, forany x ∈ ∆ o e ( S/ [ n ]) \ ∆ o e ( S/ [ n ]) l , the functor µ l induces an equivalence∆ o e ( S l ) ∼ = ∆ o e ( S/ [ n ]) l / π ∗ R x , and ∆ o e ( S/ [ n ]) l / π ∗ R x ⊂ ∆ o e ( S/ [ n ]) l /x is left-admissible by Example 1.6, so that we obtain a collection of functors(8.20) ε ( x ) : ∆ o e ( S l ) → ∆ o e ( S/ [ n ]) , x ∈ ∆ o e ( S/ [ n ]) \ ∆ o e ( S/ [ n ]) l equipped with admissible augmentations ε ( x ) > sending o to x . By Re-mark 1.13, this gives a framing for the embedding ∆ o e ( S/ [ n ]) l ⊂ ∆ o e ( S/ [ n ]).130oreover, if x ∆ o e ( S/ [ n ]) l , then ∆ o e ( S/ [ n ]) l /x = ∆ o e ( S/ [ n ]) l /x , so thatwe also obtain a framing for the embedding β . Then by Lemma 1.14 (i) and(2.4), we have(8.21) β ∆ o ! A ( h [ m ] , e q i ) = colim ∆ o e ( S l ) A ε ( h [ m ] , e q i ) , for any h [ m ] , e q i ∈ ∆ o e ( S/ [ n ]) \ ∆ o e ( S/ [ n ]) l , where we denote(8.22) A ε ( h [ n ] , e q i = ζ ([ m ]) ◦ ε ( h [ m ] , e q i ) ∗ A : ∆ o e ( S l ) → B C [ m ] , and we have to prove that the colimits in (8.21) exist.Indeed, by (6.4), we have B C [ m ] ∼ = C m , and it suffices to prove thatcolimits in (8.21) exist after projecting to each factor C . The projections aregiven by the transition functors with respect to anchor maps a : [1] → [ m ],and since A is a lax 2-functor, we have a o ! ◦ A ε ( h [ m ] , e q i ∼ = A ε ( h [1] , a ∗ e q i ),so may assume right away that [ m ] = [1]. Moreover, the map f = π ◦ e q :[ m ] → [ n ] factors through [ n ′ ] = f ([ m ]) ∪ { l } ⊂ [ n ], and since iteratedind-polycylinders are preserved by pullbacks, we may replace [ n ] with [ n ′ ]without changing anything, so that we may further assume that [ n ] = [2]. If l = 0 or l = 2, A ε ( h [1] , e q i ) is constant, and since the simplicial set N ( e ( S l ))is contractible, there is nothing to prove. If l = 1, then by (8.14) and (8.19),we have A ∗ ε ( h [1] , e q i ) ∼ = M s ⊠ A l N, where N is a left h S l , A l i -module, s = e , and M s is the right h S l , A l i -moduleof Lemma 8.21 (ii) for h S ≤ , A ≤ i . Then the colimit exists by Lemma 8.10.For (iii), by Lemma 1.14 (ii), we have to check that for any h [ m ] , e q i inthe complement ∆ o e ( S/ [ n ]) \ ∆ o e ( S/ [ n ]) l , the augmented functor(8.23) A ε ( h [ m ] , e q i ) = ζ ([ m ]) ◦ ε ( h [ m ] , e q i ) ∗ > A : ∆ o> e ( S l ) → B C [ m ] is exact. It suffices to prove that it is a filtered colimit of functors contractiblein the sense of Definition 3.1. We again can project to components of thedecomposition B C [ m ] ∼ = C m , and all projections but one are constant, thusextend to any contraction of N ( e ( S l )). Thus as in the proof of (ii), we mayassume that [ m ] = [1], [ n ] = [2] and l = 1. Then if we again let s = e , theembedding ε ( h [1] , e q i ) factors through the embedding j s : ∆ o e ( S ≥ / [1]) → ∆ o e ( S/ [2]) of (8.14), and by Lemma 8.21 (ii), the ∆ o e ( S ≥ / [1])-module M s isind-polyrepresentable. If M ∼ = V s ′ , V ∈ C , s ′ ∈ S is polyrepresentable, thenby (8.5) and (8.14), ζ ([1]) ◦ j ∗ s A extends to the category ∆ o + e ( S ≥ / [1]) s ′ , andthen A ε ( h [1] , e q i ) extends to ∆ o + e ( S ) s ′ , so that it is a contractible functor.In the general case, it is a filtered colimit of such. (cid:3) orollary 8.23. Assume given an iterated ind-polycylinder h S/ [ n ] , A i , andlet ∆ op e ( S/ [ n ]) ⊂ ∆ o + e ( S/ [ n ]) be the opposite categories to the categories of (3.20) , with the embedding functor β : ∆ op e ( S/ [ n ]) → ∆ o + e ( S/ [ n ]) . Then theadjunction map β ∆ o ! β ∗ A → A is an isomorphism.Proof. Use the same induction as in the first claim of Lemma 3.9, andreplace its second claim with Lemma 8.22. (cid:3) Proof of Proposition 8.20. Since all the cylinder conditions of Definition 8.16are stable by pullbacks, all the subcategories in question are subcofibra-tions by Lemma 2.11 (i). Their fibers over [0] ∈ ∆ o are discrete, so itsuffices to check the Segal condition. For any square (3.6), any triple ofaugmented sets S ≤ l / [ l ], S ≥ l / [ n − l ] and a set S l equipped with isomorphisms S l ∼ = S ≤ ll ∼ = S ≥ l define an [ n ]-augmented set S = S ≤ l ⊔ S l S ≥ l . Then lax2-functors A ≤ l : ∆ o e ( S ≤ l / [ l ]), A ≥ l : ∆ o e ( S ≥ l / [ l ]) together with an isomor-phism between their restrictions to the subcategory (8.16) define a lax 2-functor A : ∆ o e ( S/ [ n ]) l → B C that uniquely extends to a lax 2-functor A : ∆ o e ( S/ [ n ]) l → B C by Lemma 8.22 (i). We have to shows that if h S ≤ l / [ l ] , A ≤ l i and h S ≥ l / [ l ] , A ≥ l i are iterated cylinders resp. polycylindersresp. ind-cylinders resp. ind-polycylinders, then A uniquely extends to alax 2-functor A : ∆ o e ( S/ [ n ]) → B C such that h S/ [ n ] , A i is an iterated cylin-der resp. polycylinder resp. ind-cylinder resp. ind-polycylinder.To do this, we note that firstly, uniqueness follows from Lemma 8.22,and secondly, by induction, it suffices to consider the case l = 1. Then byLemma 8.21, it suffices to assume that h S ≤ / [1] , A ≤ i is a cylinder resp.polycylinder resp. ind-cylinder resp. ind-polycylnder, take the lax 2-functor A = β ∆ o ! A provided by Lemma 8.22 (ii), and prove that for any s ∈ S ,the h S ≥ l / [ n − , A ≥ i -module M s of (8.14) is representable resp. polyrepre-sentable resp. ind-representable resp. ind-polyrepresentable.Indeed, to simplify notation, let S ′ = S ≥ , A ′ = A ≥ , so that S ′ ∼ = S ,and consider the subcategory∆ o e ( S ′ / [ n − = ∆ o e ( S ′ / [ n − ∩ ∆ o e ( S/ [ n ]) l ⊂ ∆ o e ( S ′ / [ n − , with the embedding functor β : ∆ o e ( S ′ / [ n − → ∆ o e ( S ′ / [ n − β , and the functor j s of (8.13) intertwines β with the embedding β of (8.17) and identifies the framings, so that we have j ∗ s ◦ β ∆ o ! ∼ = β ∆ o ! ◦ j ∗ s . Therefore by (8.14) and (8.19), the h S ′ / [ n − , A ′ i -module M s is given by(8.24) M s = β ∆ o ! ( µ ∗ ( M ≤ s ) × µ ∗ A A ′ ) , µ : ∆ o> e ( S ′ / [ n − → ∆ o> e ( S ) is left-adjoint tothe embedding ∆ o> e ( S ) ∼ = ∆ o> e ( S ′ ) → ∆ o> e ( S ′ / [ n − M ≤ s is the h S , A i -module of Lemma 8.21 for the enriched category h S ≤ / [1] , A ≤ i .If this category is a polycylinder, then M ≤ s ∼ = V s ′ ∼ = ρ o ( s ′ ) ∆ o ! V + s for some V ∈ C , s ′ ∈ S ∼ = S ′ , and then by the same argument as in Lemma 8.10,(8.24) yields M s ∼ = ρ o ( s ′ ) ∆ o ! V + s ′ ∼ = V s ′ ∈ A ≥ -mod , where s ′ is now considered as an initial object in e ( S ′ / [ n − M s is polyrepresentable as required. If h S ≤ / [1] , A ≤ i is a cylinder, then V ∼ = 1,so M s is representable, and in the ind-cases, note that left Kan extensionscommute with filtered colimits. (cid:3) Let us now describe the relationship be-tween the 2-categories of Proposition 8.20 and the category Cat( C ) of small C -enriched categories and functors between them.Assume given C -enriched categories h S , A i , h S , A i and a functor h f, g i : h S , A i → h S , A i , and let S = S ⊔ S , with the projection S → [1] sending S l to l , l = 0 , 1. Then the cylinder C ( e ( f )) of the func-tor e ( f ) : e ( S ) → e ( S ) is naturally identified with e ( S / [1]), and the2-cylinder C ( e ( f ) , g ) of Lemma 6.16 defines a [1]-augmented C -enrichedcategory(8.25) C ( f, g ) = h S / [1] , C ( e ( f ) , g ) i . By (6.22) and (8.14), for any s ∈ S , the h S , A i -module M s of Lemma 8.21corresponding to (8.25) is isomorphic to the representable module 1 f ( s ) , sothat C ( f, g ) is a cylinder in the sense of Definition 8.16. This suggests thatthere exists a 2-functor(8.26) ∆ o Cat( C ) → C at ( C )sending h S, A i to itself and h f, g i to C ( f, g ).To construct such a 2-functor, let S : Cat( C ) → Sets be the forgetfulfunctor sending h S, A i to S , let Cat q ( C ) → Cat( C ) be the cofibration withfibers e ( S ), and let Cat q ∆ ( C ) → Cat( C ) be the cofibration with fibers ∆ o e ( S )(equivalently, Cat q ∆ ( C ) → Cat( C ) × ∆ o is the discrete cofibration correspond-ing to ε ∗ S : Cat( C ) × ∆ o → Sets, where ε : Cat( C ) → Cat( C ) × ∆ o is the em-bedding onto Cat( C ) × [0]). Then the functors A for various h S, A i ∈ Cat( C )together define a single functor A : Cat q ∆ ( C ) → B C over ∆ o that is cocarte-sian over anchor maps. On the other hand, the functors (6.10) together133efine a functor ξ : Cat q ∆ ( C ) → Cat q ( C ) cocartesian over Cat( C ), and thenas in Lemma 6.16, (2.24) converts it to a functor(8.27) α : Cat q ∆ ( C ) → Id + ∗∗ (Cat q × ∆ o ) = ∆ o h Cat q i over ∆ o . Lemma 8.24. The right Kan extension α ∗ A with respect to (8.27) exists,its restriction to ∆ o Cat[ S ] = ∆ o Cat q ⊂ ∆ o h Cat q i defines a C -enrichementof the -category ∆ o Cat , and the corresponding functor (8.9) factors througha -functor (8.26) .Proof. Same as Lemma 6.16. (cid:3) Since an iterated cylinder is tautologically an iterated polycylinder, wehave C at ( C ) ⊂ M or ( C ) ⊂ M or ∗ ( C ), so that (8.26) also defines a 2-functorfrom Cat( C ) to M or ( C ) and M or ∗ ( C ). One difference between C at ( C ) andthe Morita 2-categories is that in the latter, many morphisms become re-flexive. Namely, recall that for any C -entiched category h S, A i , we have theopposite C ι -enriched category h S, A i o = h S, A ι i , and note that a functor h f, g i : h S , A i → h S , A i defines an opposite functor h f, g ι i between theopposite categories. Then (8.25) gives an [1]-augmented C ι -enriched cate-gory C ( f, g ι ), and if we let S / [1] be S whose augmentation map S → [1]is composed with the order-reversing isomorphism [1] o ∼ = [1], then(8.28) C ι ( f, g ) = C ( f, g ι ) ι = h S / [1] , C ( e ( f ) , g ι ) i is a [1]-augmented C -enriched category that we call the dual cylinder of thefunctor h f, g i . Definition 8.25. A functor h f, g i : h S , A i → h S , A i between small C -enriched categories is reflexive resp. ind-reflexive if its dual cylinder (8.28)is a polycylinder resp. an ind-polycylinder. Example 8.26. By Example 8.19, the functors i s of Example 8.3 are alwaysreflexive in the sense of Definition 8.25. Example 8.27. If A = k -mod, as in Example 8.9, and S = S = pt , sothat A and A are k -algebras and g : A → A is an algebra map, then g is reflexive resp. ind-reflexive if A is free resp. flat as a module over A .134 roposition 8.28. Assume given a functor h f, g i : h S , A i → h S , A i between small C -enriched categories that is reflexive resp. ind-reflexive in thesense of Definition 8.25. Then the morphism C ( f, g ) is reflexive in M or ( C ) resp. M or ∗ ( C ) . Before proving Proposition 8.28 in general, let us consider the situationof Example 8.26: we have a small C -enriched category h S, A i and an object s ∈ S , and we want to extend the functor i s : pt C → h S, A i to an adjointpair(8.29) γ s : adj → M or ( C ) . We start by recalling the description of the 2-category eq in terms of in-jective maps a : [ n ] → [ n ]. For any such map a , the pullback functor a ∗ : Sets / [ n ] → Sets / [ n ] has a fully faithful right-adjoint a ∗ : Sets / [ n ] → Sets / [ n ] sending S ∈ Sets / [ n ] to a ∗ S ∈ Sets / [ n ] with ( a ∗ S ) i = S i for i ∈ [ n ] and ( a ∗ S ) i = pt for i ∈ [ n ] . For any S ∈ Sets / [ n ], we then havethe [ n ]-augmented set S a = a ∗ a ∗ S and the adjunction map p : S → S a .Moreover, let ∆ o e ( S a / [ n ]) ⊂ ∆ o e ( S a / [ n ]) be the full subcategory spannedby objects h [ m ] , s i such that the composition [ m ] → S a → [ n ] is 0-special.Then by virtue of (7.17), the embedding δ : ∆ o e ( S a / [ n ]) → ∆ o e ( S a / [ n ])admits a right-adjoint functor δ † : ∆ o e ( S a / [ n ]) → ∆ o e ( S a / [ n ]). Thereforefor any lax 2-functor A : ∆ o e ( S a / [ n ]) → B C , the Kan extension(8.30) A a = δ ∆ o ! δ ∗ A : ∆ o e ( S a / [ n ]) → B C exists and is given by (2.6), so that it is a lax 2-functor. In particular,if we have an [ n ]-augmented C -enriched category h S/ [ n ] , A i and a section i : S a → S of the adjunction map p : S → S a , then A i = ∆ o e ( i ) ∗ A is a lax2-functor, and we have the [ n ]-augmented C -enriched category h S a / [ n ] , A i,a i . Lemma 8.29. In the assumptions above, if h S/ [ n ] , A i is an iterated poly-cylinder in the sense of Definition 8.16, then so is h S a / [ n ] , A i,a i .Proof. If n = 1, the claim immediately follows from Example 8.19. If n ≥ l ∈ [ n ], 0 < l < n , consider the square (3.6), let[ l ] = a ([ n ] ) ∩ s ([ l ]) ⊂ [ n ] , [ n − l ] = a ([ n ] ) ∩ t ([ n − l ]) ⊂ [ n ] , with the embedding maps a ′ : [ l ] → [ l ], a ′′ : [ n − l ] → [ n − l ], and let S ′ = s ∗ S , S ′′ = t ∗ S , with the induced maps i ′ : S ′ a ′ → S ′ , i ′′ : S ′′ a ′′ → S ′′ . Then if welet A ′ = s ∗ A , A ′′ = t ∗ A , we have s ∗ h S a / [ n ] , A i,a i ∼ = h S ′ a ′ / [1] , A ′ i ′ ,a ′ i , t ∗ h S a / [ n ] , A i,a i ∼ = h S ′′ a ′′ / [ n − , A ′′ i ′′ ,a ′′ i C -enriched categories are iterated poly-cylinders by induction. As in the proof of Proposition 8.20, they define aniterated polycylinder h S a , A ′ i , and we need to check that A i,a ∼ = A ′ . More-over, if we consider the factorization (8.17) for the category ∆ o e ( S a / [ n ]),then A i,a ∼ = A ′ on ∆ o e ( S a / [ n ]) l , and hence also on ∆ o e ( S a / [ n ]) l by virtueof Lemma 8.22 (i). Thus by Lemma 8.22 (iii), it suffices to check that theadjunction map β ∆ o ! β ∗ A i,a → A i,a is an isomorphism. To do this, use theframing (8.20) for β , and note that if l ∈ [ n ] \ [ n ] , then S a,l ∼ = S l , and ∆ o e ( i )identifies the framings, so that δ ∆ o ! δ ∗ ∆ o e ( i ) ∗ commutes with β ∆ o ! , and we arethen done by Lemma 8.22 (iii). On the other hand, if l ∈ [ n ] , then S a,l = pt is the single point, and for any h [ m ] , e q i ∈ ∆ o e ( S a / [ n ]) \ ∆ o e ( S a / [ n ]) l , theaugmented functor (8.23) is constant, hence exact. (cid:3) We can now construct the adjoint pair (8.29) for any small C -enrichedcategory h S, A i and element s ∈ S . To do this, let S : adj [0] = eq [0] → Setsbe the constant functor with value S , let S + : eq [0] = { , } → Sets bethe functor sending 1 to S and 0 to the point, and let i + s : S + → S bethe map equal to id at 1 and to the embedding i s : pt → S onto s at0, with the corresponding 2-functor ι s : eq [ S + ] → eq [ S ]. Moreover, let η : eq [ S + ] → adj [ S + ] be the lax 2-functor induced by (7.11). Since as wesaw in the proof of Lemma 7.4, η admits a right-adjoint, the Kan extension η ∆ o ! E exists for any functor E : eq [ S + ] → B C over ∆ o , and is given by (2.6).In particular, we can take the constant 2-functor eq → M or ( C ) with value h S, A i , with the corresponding constant C -enrichment E : eq [ S ] → B C ofthe 2-category eq , and consider the functor(8.31) E + = η ∆ o ! ι ∗ s E : adj [ S + ] → B C . Then (2.6) immediately shows that this is a lax 2-functor, so that h S + , E + i isa C -enrichment of the 2-category adj . Moreover, for any object c ∈ eq ⊂ adj represented by an injective map a : [ n ] → [ n ], the right comma-fiber c \ adj of the cofibration adj [ S + ] → adj is identified with ∆ o e (( S × [ n ]) a / [ n ]), andthe projections (1.1) for the cofibrations eq [ S + ] → eq , adj [ S + ] → adj fit intoa cartesian diagram(8.32) ∆ o e (( S × [ n ]) a ) δ −−−−→ ∆ o e (( S × [ n ]) a ) p c y y p c eq [ S + ] η −−−−→ adj [ S + ] . Then (2.6) shows that the base change map δ ∆ o ! ◦ p ∗ c → p ∗ c ◦ η ∆ o ! is anisomorphism. Therefore the functor Y ( S + , E + ) of (8.9) sends the object c eq ⊂ adj to the C -enriched [ n ]-augmented category h ( S × [ n ]) a / [ n ] , ( t ( c ) ∗ ι ∗ s E ) a i . By Lemma 8.29, this is an iterated polycylinder,so that Y ( S + , E + ) sends eq ⊂ adj into M or ( C ) ⊂ Aug( C ). Since Y ( S + , E + )is cocartesian over ∆ o , it then sends the whole adj into M or ( C ) ⊂ Aug( C ),and gives the adjoint pair (8.29). In order to prove Proposition 8.28 in full generality, weneed two technical results. The first is an enriched version of Lemma 6.15.Assume given a quiver Q , with the path category P ( Q ). Recall that thesimplicial replacement ∆ o P ( Q ) is given by (6.18), and let D op P ( Q ) = ar p (∆) o♮ × ∆ oa D o Q ⊂ ∆ op P ( Q ) = ar p (∆) o♮ × ∆ oa ∆ oa A ( Q ) ⊂ ∆ o P ( Q ) . Explicitly, objects in ∆ o P ( Q ) are pairs h f, x i of a bispecial arrow f : [ n ] → [ m ] in ∆ and a quiver map x : [ m ] δ → Q , ∆ op P ( Q ) ⊂ ∆ o P ( Q ) is spanned bypairs with surjective f , and D op P ( Q ) ⊂ ∆ op P ( Q ) is spanned by pairs with m ≤ 1. Then for any C -enrichment h S, A i of ∆ o P ( Q ), the full embeddings D op P ( Q ) ⊂ ∆ op P ( Q ) ⊂ ∆ o P ( Q ) give rise to full embeddings(8.33) D op P ( Q )[ S ] α −−−−→ ∆ op P ( Q )[ S ] β −−−−→ ∆ o P ( Q )[ S ]over ∆ o , and A induces functors A p = β ∗ A : ∆ op P ( Q )[ S ] → B C and A d = α ∗ A p : D op P ( Q )[ S ] → B C , again over ∆ o . Lemma 8.30. In the assumptions above, assume further that the functor Y ( S, A ) of (8.9) takes values in M or ∗ ( C ) ⊂ Aug( C ) . Then the adjunctionmaps A p → α ∗ A d and A → β ∆ o ! A p are isomorphisms (and both Kan exten-sions exist).Proof. The projection (6.19) restricts to a functor t : ∆ o P ( Q )[ S ] → ∆ oa A ( Q ),and by (3.15), this functor is a cofibration. Then a framing for the functor α o is obtained by taking cocartesian liftings of the right comma-fibers ofthe functor α ( Q ) : D o Q → ∆ oa A ( Q ) of (6.16), and then as in the proof ofLemma 6.15, computing α ∗ by means of this framing amounts to takingiterated limits of standard squares in B C , so that α ∗ A d exists and A p ∼ = α ∗ A d . As for the Kan extension β ∆ o ! , note that β is a functor over ∆ oa A ( Q ),and a framing for β is given by left comma-fibers of its fibers β x , x ∈ ∆ oa A ( X ). But every such fiber is an embedding β of Corollary 8.23. (cid:3) The second result is a corollary of Lemma 3.10. Consider the adjunction2-category adj , let adj p ⊂ adj be the full subcategory spanned by diagrams1377.5) with surjective map b , and let adj p♮ = adj p ∩ adj ♮ . We then have 2-categories adj p♮ ⊂ adj p , adj ♮ ⊂ adj with the same objects 0, 1, and for any S : { , } → Sets, we have a commutative diagram(8.34) adj p♮ [ S ] δ −−−−→ adj ♮ [ S ] γ y y γ adj p [ S ] δ −−−−→ adj [ S ] , where all the arrows are fully faithful lax 2-functors Lemma 8.31. For any C -enrichment h S, A i of the -category adj p , the basechange map δ ∆ o ! γ ∗ A → γ ∗ δ ∆ o ! A induced by (8.34) is an isomorphism, andits target exists iff so does its source.Proof. Let P p ( eq ) ⊂ P ( eq ) be the full subcategory spanned by surjectivebispecial arrows, and consider the diagram(8.35) P p ( eq )[ S ] t ∗ a δ −−−−→ P ( eq )[ S ] t ∗ a γ y y γ P p ( eq )[ S ] δ −−−−→ P ( eq )[ S ] , where a A ′ = l ∗ A , then A ∼ = l ∆ o ! A ′ . Therefore it suffices to prove the claim for the base change mapassociated to (8.35). To do this, it suffices to prove that for any c ∈ P ( eq )[ S ],the embedding(8.36) P p ( eq )[ S ] / δt ∗ a c → P p ( eq )[ S ] / δ c induced by δ : P p ( eq ) → P ( eq ) is cofinal. By Lemma 1.18, it suffices toprove this with a Iso . But then, the projection P ( eq )[ S ] → P ( eq ) → P ( pt ) = ar ± (∆) o is a discrete cofibration, and it identifies theright comma-fibers of the embedding (8.36) with those of (3.22), so we aredone by Lemma 3.10. (cid:3) Proof of Proposition 8.28. By Remark 7.6, the 2-category adj ♮ ∼ = ∆ o P ([2] λ )is the simplicial replacement of the path category of the quiver [2] λ , so thatLemma 3.4 provides a canonical 2-functor adj ♮ → M or ∗ ( C ) — or to M or ( C )138n the reflexive case — that sends 0 resp. 1 to h S , A i resp. h S , A i , andthe two edges of the quiver to C ( f, g ) and C ι ( f, g ). Let h S, A i be the corre-sponding C -enrichment of the 2-category adj ♮ . Then it suffices to prove that A : adj ♮ [ S ] → B C extends to adj [ S ] ⊃ adj ♮ [ S ]. Moreover, by Lemma 8.30, wehave A ∼ = β ∆ o ! A p , and the embedding β of (8.33) factors as∆ op P ([2] λ ) ε −−−−→ adj p♮ [ S ] δ −−−−→ adj ♮ [ S ] ∼ = ∆ o P ([2] λ )[ S ] , where δ is as in (8.34). Therefore by Lemma 8.31, it suffices to find a functor A w : adj p [ S ] → B C over ∆ o and an isomorphism γ ∗ A w ∼ = ε ∆ o ! A p — then δ ∆ o ! A w exists and restricts to A on adj ♮ [ S ] ⊂ adj [ S ].To construct A w , as in Remark 6.17, decompose the map f : S → S as(8.37) S σ −−−−→ S ζ −−−−→ S , where σ : S → S ⊔ S = S is the embedding, and the map ζ is equal to f on S ⊂ S and to id on S ⊂ S . Denote h S , A i = ζ ∗ h S , A i . Thenwe have ∆ o e ( σ ) ∗ A ∼ = ∆ o e ( f ) ∗ A , and the functor h f, g i factors through afunctor h σ, g i : h S , A i → h S , A i . Since σ is injective, ∆ o e ( σ ) is fullyfaithful, and the cylinder C (∆ o e ( σ )) is a full subcategory in ∆ o e ( S ) × [1].Then the functor w of (6.32) restricts to a functor(8.38) w : C (∆ o e ( σ )) → adj p [ S ] ⊂ B [1][ S ] , while the functor h σ, g i defines a functor A g : C (∆ o e ( σ )) → B C over ∆ o given by A resp. A over 0 × ∆ o resp. 1 × ∆ o . The same argument as inLemma 6.25 then shows that the right Kan extension(8.39) A w = w ∗ A g : adj p [ S ] → B C exists and is a lax 2-functor.To construct the isomorphism, consider the restriction α ∗ ε ∗ γ ∗ A w of thefunctor A w to D op P ([2] λ )[ S ]. Then to obtain an isomorphism α ∗ ε ∗ γ ∗ A w ∼ = A d , it suffices to observe that∆ o e ( S ) ⊔ ∆ o e ( S ) ∼ = ∆ o e ( S / [1]) ∩ ∆ o e ( S / [1]) , D op P ([2] λ )[ S ] ∼ = ∆ o e ( S / [1]) ∪ ∆ o e ( S / [1]) , apply Example 3.3, and interpret the cylinder and the dual cylinder in termsof (6.25). Moreover, we then get the adjoint map ε ∗ γ ∗ A w → A p ∼ = α ∗ A d between lax 2-functors, and since lax 2-functors preserve limits of standardsquares, the same argument as in Lemma 8.30 shows that this is also an139somorphism. Thus by Lemma 8.30, it suffices to prove that the adjunctionmap(8.40) ε ∆ o ! ε ∗ γ ∗ A w → γ ∗ A w is an isomorphism. This is again a map of lax 2-functors, so it further sufficesto prove that it is an isomorphism on the fiber adj p [ S ] [1] over [1] ∈ ∆ o .Now note that the whole fiber adj p [ S ] [1] lies in the essential image of thefunctor w of (8.38) (this amounts to observing that a surjective bispecial map[ n ] → [ m ] in ∆ with [ n ] = [1] is either a bijection, or the projection onto[0]). Therefore it suffices to prove that (8.40) becomes an isomorphism afterapplying w ∗ . Moreover, w sends ∆ o e ( S ) = C (∆ o e ( σ )) into ∆ op P ([2] λ )[ S ] ⊂ adj p♮ [ S ] where (8.40) is tautologically an isomorphism, so it suffices to furtherrestrict to C (∆ o e ( σ )) = ∆ o e ( S ) = eq [ S ]. The corresponding component w = η : eq [ S ] → adj p [ S ] of the functor w is then the embedding (7.11). Itis fully faithful, and we have(8.41) η ∗ A w ∼ = A ∼ = ν ∆ o ! A w , where ν : adj p [ S ] → eq [ S ] is left-adjoint to η induced by (7.4). Moreover, eq [ S ] ∩ adj p♮ [ S ] ⊂ eq [ S ] is the dense subcategory eq [ S ] a , the functors η and ν induce an adjoint pair of functors between eq [ S ] a and adj p♮ [ S ], and if welet γ : eq [ S ] a → eq [ S ] be the embedding, then (8.41) also holds for therestrictions γ ∗ A w and γ ∗ A . Finally, if we denote eq p [ S ] a = eq [ S ] a ∩ ∆ op P ([2] λ )[ S ], with the embedding ε : eq p [ S ] a → eq [ S ] a , then η and ν further induce an adjoint pair of functors between eq p [ S ] a and ∆ op P ([2] λ )[ S ],(8.41) still holds, and η ∗ sends the map (8.40) to the adjunction map(8.42) ε ∆ o ! ε ∗ γ ∗ A → γ ∗ A . But A only depends on f : S → S and A , so that from now on, we mayassume that g : A → ∆ o e ( f ) ∗ A is an isomorphism. Moreover, if we let S ′ = S ′ = S and A ′ = A ′ = A , then the map ζ of (8.37) factors through amap h = f ⊔ id : S → S ′ = S ′ ⊔ S ′ , and the functor h : eq [ S ] a → eq [ S ′ ] a identifies the left comma-fibers of the embeddings ε : eq p [ S ] a → eq [ S ] a and ε : eq p [ S ′ ] a → eq [ S ′ ] a , so that h ∗ ◦ ε ∆ o ! ∼ = ε ∆ o ! ◦ h ∗ . Therefore the map (8.42)is the pullback of the same map for the functor h id , id i : h S , A i → h S , A i ,and we may assume right away that h f, g i = h id , id i . But then A w triviallyextends to the whole adj [ S ], and (8.40), hence also (8.42) is an isomorphismby Lemma 8.30. (cid:3) .6 Admissible bases. It turns out that the adjoint pairs (8.29) allowone to characterize the Morita 2-category by a universal property refiningLemma 8.15. To do this, it is convenient to change notation: assume given a2-category C , and assume that C is pointed , in the sense that we have a dis-tinguished object o ∈ C [0] . Let C o = C ( o, o ) be the endomorphisms categoryof the object o , with its natural unital monoidal structure of Example 6.19. Definition 8.32. A base of a pointed 2-category hC , o i is a collection of non-empty sets S ( c ), c ∈ C [0] of reflexive morphisms i s ∈ C ( o, c ), s ∈ S ( c ), suchthat s ( o ) consists of the unit object 1 ∈ C o = C ( o, o ). A base is admissible if for any c, c ′ ∈ C [0] , f ∈ C ( c, c ′ ) and s ∈ S ( c ) we have f ◦ i s ∼ = i s ′ ◦ V forsome s ′ ∈ S ( c ′ ), V ∈ C o , and strongly admissible if one can arrange that V ∼ = 1. If C has filtered colimits, then a base is ind-admissible resp. stronglyind-admissible if for any c, c ′ ∈ C [0] , f ∈ C ( c, c ′ ) and s ∈ S ( c ), f ◦ i s is afiltered colimit of objects of the form i s ′ ◦ V resp. i s ′ .For any pointed 2-category hC , o i with a base S , the adjoint pairs (8.29)taken together define an adjoint pair(8.43) γ : C [ S ] [0] × adj → C . Since C [ S ] [0] is discrete, γ is also a coadjoint pair, so that by Proposition 7.9,it gives rise to the twisting lax 2-functor Θ( γ ) : γ ∗ C → γ ∗ C of (7.39). Notethat we have γ ∗ C ∼ = C [ S ], while γ ∗ C ∼ = C [ S ] [0] × B C o is the disjoint unionof several copies of the category B C o . If we let τ : C [ S ] [0] → pt be thetautological projection, then we have the lax 2-functor(8.44) Θ = ( τ ◦ id ) ◦ Θ( γ ) : C [ S ] → B C o , and the pair h S, Θ i is a C o -enrichment of the 2-category C in the sense ofDefinition 8.13. Lemma 8.33. Assume that the base S of a pointed -category C is admis-sible resp. strongly admissible. Then the functor Y = Y ( S, Θ) of (8.9) corre-sponding to the enrichment (8.44) factors through M or ( C o ) resp. C at ( C o ) ⊂ Aug( C o ) . Moreover, assume that C has filtered colimits. Then if S is ind-admissible resp. strongly ind-admissible, Y factors through M or ∗ ( C o ) resp. C at ∗ ( C o ) .Proof. Note that the description of right A -modules in terms of 1-speciallax 2-functors (8.6) also works in in the augmented setting: for any [ n ] ∈ ∆and C o -enriched [ n ]-augmented category h S/ [ n ] , A i , we have e ( S/ [ n ]) < ∼ =141 ( S + / [ n +1]), with the augmentation map S + → [ n +1] = [ n ] < sending S to[ n ] = t ([ n ]) ⊂ [ n +1] and o ∈ S + to 0, and then right A -modules M corre-spond bijectively to lax 2-functors M † : ∆ o e ( S + / [ n +1]) → B C o , equippedwith an isomorphism M † | ∆ o e ( S/ [ n ]) ∼ = A and 1-special with respect to theprojection S + → { , } sending o to 0 and S to 1. Now take some object c ∈ C [ n ] ⊂ C with b ( c ) = c ∈ C [0] , an element s ∈ S ( c ) ⊂ S ( c ) withthe corresponding reflexive morphism i s ∈ C ( o, c ), and an object V ∈ C o .Denote V s = i s ◦ V ∈ C ( o, c ) ⊂ C [1] , and consider the product V s × c ∈ C [ n +1] ∼ = C [1] × C [0] C [ n ] . Then since S ( o ) consists of the single element 1 ∈ C o , we have an identifica-tion S ( V s × c ) ∼ = S ( c ) + , and Y ( V s × c ) : ∆ o e ( S ( c ) + / [ n +1]) → B C o is 1-specialby virtue of (7.40), thus corresponds to a right module over Y ( c ). Moreover,again by (7.40), the functor (8.1) sends this module to i ∨ s ◦ V s ∈ C o , theadjunction map V → i ∨ s ◦ i s ◦ V ∼ = i ∨ s ◦ V s induces a map V † s → Y ( V s × c ) , where V † s corresponds to the polyrepresentable right Y ( c )-module V s , and(7.40) immediately implies that this map is an isomorphism. This finishesthe proof: since Y is cocartesian over ∆ o , we have f ! Y ( V s × c ) ∼ = Y ( f ! ( V s × c )),and then the four cylinder conditions of Definition 8.16 directly translate tothe corresponding admissibility conditions of Definition 8.32. (cid:3) The 2-category M or ∗ ( C o ) itself is pointed by the point enriched category o = pt C o of Example 8.3, with M or ∗ ( C o ) o ∼ = C o by (8.12), and it is equippedwith the standard base S triv given by all the reflexive maps i s of (8.29).This base is ind-admissible by definition, and its restriction to M or ( C o ) ⊂M or ∗ ( C o ) is admissible. Let us now show that the base S triv enjoys auniversal property analogous to Lemma 8.15. Proposition 8.34. For any pointed -category hC , o i equipped with an ad-missible base S , there exists a -functor (8.45) Y : C → M or ( C o ) equipped with an isomorphism ϕ : S ∼ = Y ∗ S triv and isomorphisms i ϕ ( s ) ∼ = Y ( i s ) for any s ∈ S ( c ) , c ∈ C [0] . Moreover, such a functor is unique up to aunique isomorphisms, and if C has filtered colimits, then the same statementholds for an ind-admissible base S and the -category M or ∗ ( C o ) . emark 8.35. The 2-functor (8.45) is a sort of a 2-Yoneda embedding forthe pointed 2-category C : we realize objects c ∈ C [0] by C o -enriched categoriesformed by objects i s ∈ C ( o, c ). This motivates our notation. Proof. By virtue of Lemma 8.33, it suffices to consider the universal sit-uation: we take the C o -enrichment Θ triv of the 2-category M or ( C o ) cor-responding to the base S triv , and we need to construct an isomorphism Y ( S triv , Θ triv ) ∼ = Id (the same argument will also work for M or ∗ ( C o ) in theind-admissible case). By Lemma 8.15, it further suffices to construct anisomorphism(8.46) Θ triv ∼ = ev , where ev : M or ( C o )[ S triv ] → B C o is induced by the evaluation functor (8.8).To do this, we construct explicitly the decomposition (7.32) for the disjointunion(8.47) γ : M or ( C o )[ S triv ] [0] × adj → M or ( C o )of the adjoint-coadjoint pairs (8.29). We again use Lemma 8.29. To sim-plify notation, denote C = M or ( C o )[ S triv ] ∼ = γ ∗ M or ( C o ), with the naturalprojection π : C{ eq } → C = M or ( C o )[ S triv ] → M or ( C o ), and denote also S = π ∗ S triv : C{ eq } [0] = C [0] × eq [0] → Sets. Moreover, let S + be the func-tor from C{ eq } [0] to sets equal to S over 0 ∈ { , } = eq [0] , and sending C [0] × i + : S + → S be equal to id over C [0] × i s on h S/ [ n ] , A, s i × ∈ C [0] × eq [0] , with the corre-sponding induced 2-functor ι : C{ eq } [ S + ] → C{ eq } [ S ]. Furthermore, denoteby η : C{ eq } [ S + ] → C{ adj } [ S + ] the lax 2-functor induced by (7.12), andlet E : C{ eq } [ S ] → B C o be the C o -enrichment of the 2-category C{ eq } [ S ]corresponding to the projection π . Finally, let E + = η ∆ o ! ι ∗ E : C{ adj } [ S + ] → B C o , where as in (8.31), the Kan extension exists by (2.6), and defines a lax 2-functor, so that h S + , E + i is a C o -enrichment of the 2-category C × adj . More-over, for any object in eq represented by an injective arrow a : [ n ] → [ n ],and any object h S/ [ n ] , A, s i ∈ C [ n ] , we have a counterpart of the cartesiandiagram (8.32), and we conclude by base change that Y ( S + , E + ) sends theproduct h S/ [ n ] , A, s i × a ∈ C × ∆ o eq to the C -enriched [ n ]-augmented cat-egory h S a / [ n ] , ( i ∗ a,s A ) a i , where i a,s : S a → S is equal to id over [ n ] , andsends the single element in ( S a ) l , l ∈ [ n ] to the value s ( l ) ∈ S l of the sec-tion s : [ n ] → S . Then by Lemma 8.29, this is an iterated polycylinder, so143hat Y ( S + , E + ) : C{ adj } → Aug( C o ) factors through M or ( C o ) ⊂ Aug( C o )and then restricts to a 0-special lax 2-functor(8.48) γ ⋄ : C{ eq } → M or ( C o )that fits into a decomposition (7.32) for the adjoint pair (8.47).To finish the proof in the admissible case, it remains to recall that byProposition 7.9, the decomposition (7.32) is unique. Therefore Θ triv in (8.46)is naturally identified with the component γ ⋄ : C = M or ( C o )[ S triv ] → B C o ⊂M or ( C o ) of the lax 2-functor (8.48). The latter sends h S/ [ n ] , A, s i to A ( s )and is canonically identified with ev .In the ind-admissible case, the proof is the same, with M or ( C o ) replacedby M or ∗ ( C o ), and Lemma 8.29 replaced with its obvious generalization foriterated ind-polycylinders. (cid:3) Now as in Proposition 8.20, let C be a unital monoidalcategory, and note that in general, the 2-categories of Proposition 8.20 forma diagram(8.49) C at ( C ) T −−−−→ M or ( C ) y y C at ∗ ( C ) T −−−−→ M or ∗ ( C ) , where all the 2-functors are faithful (the bottom row only exists if C hasfiltered colimits preserved by the tensor product). As it happens, under anadditional assumption on C , one can also construct a 2-functor going in theother direction, namely,(8.50) P : M or ∗ ( C ) → C at ∗ ( C ) . Here is the assumption. Definition 8.36. An object V in a unital monoidal category C is (left-)dua-lizable if it is reflexive as a morphism in the 2-category B C . The category C is well-generated if it has filtered colimits, the full subcategory D ( C ) ⊂ C spanned by dualizable objects is essentially small, and every object in C isa filtered colimit of dualizable objects. Example 8.37. For any commutative ring k , the category k -mod fl of flat k -modules with its usual tensor structure is well-generated, and the dualizableobjects are finitely generated projective k -modules.144ssume that the category C is well-generated in the sense of Defini-tion 8.36. Then informally, we want our 2-functor (8.50) to send a small C -enriched category h S, A i to the category of right A -modules of the form V s , s ∈ S , V ∈ D ( C ), with an appropriate enrichement. To achieve thisformally, we use the 2-functor Y of Proposition 8.34, but for a differentadmissible base.Choose a set D of representatives of the isomorphism classes of dual-izable objects, with the functor ϕ : D → D ( C ) ⊂ C (where D is treatedas a discrete category). For any C -enriched category h S, A i and s ∈ S , let i d × s = i s ◦ ϕ ( d ) ∈ M or ( C )( pt C , h S, A i ). This is a composition of reflexivemorphisms, thus reflexive. Now consider the 2-category M or ∗ ( C )[ { , } ] = M or ∗ ( C ) × eq , with the distinguished object o = pt C × 0, let f ∈ eq (0 , S + for the pointed 2-category M or ∗ ( C )[ { , } ] as follows: • for any C -enriched category h S, A i , the set S + ( h S, A i × 0) consistsof the maps i s , s ∈ S , and the set S + ( h S, A i × 1) consists of maps i d × s ◦ ( id × f ), s ∈ S , d ∈ D .This is obviously an ind-admissible base in the sense of Definition 8.32, sothat Proposition 8.34 provides a 2-functor(8.51) P + : M or ∗ ( C )[ { , } ] → M or ∗ ( C ) . Moreover, since C is well-generated, the base S + is strongly ind-admissibleon M or ∗ ( C ) × ⊂ M or ∗ ( C )[ { , } ], so that by Lemma 8.33, the restrictionof the 2-functor (8.51) to M or ∗ ( C ) × C at ∗ ( C ) ⊂ M or ∗ ( C )and induces a 2-functor (8.50).We tautologically have P ◦ T ∼ = Id , and one can probably show that P is fully faithful and right-adjoint to T of (8.49) in the appropriate 2-categorical sense, but we will not need it; the following trivial observationwill be sufficient for our purposes. Lemma 8.38. For any well-generated unital monoidal category C , the com-position T ◦ P : M or ∗ ( C ) → M or ∗ ( C ) of the -functors (8.50) and (8.49) isequivalent to the identity in the sense of (7.9) .Proof. An equivalence is given by the 2-functor (8.51). (cid:3) Trace theories. Consider now the cyclic category Λ formed by thecategories [ n ] Λ , n ≥ 1, and horizontal functors f : [ n ] Λ → [ m ] Λ . Denoteby r : Λ q → Λ be the fibration with fibers ∆ o [ n ] Λ and transition functors∆ o ( f † ) : ∆ o [ m ] Λ → ∆ o [ n ] Λ , where f † is as in (3.25). The structural cofi-brations Λ q [ n ] = ∆ o [ n ] Λ → ∆ o , [ n ] ∈ Λ then define a functor l : Λ q → ∆ o cartesian over Λ. Definition 9.1. The cyclic nerve Λ C of a 2-category C / ∆ o is the cofibrationΛ C = r ∗ l ∗ C . The cyclic nerve C ♯ of a unital monoidal category C is the cyclicnerve C ♯ = Λ B C of the corresponding 2-category B C .By definition, the cofibration Λ C → Λ has fibers Λ C [ n ] = Fun ([ n ] Λ , C ),with transition functors f ! = f ∗† : Fun ([ n ] Λ , C ) → Fun ([ m ] Λ , C ). To seethese fibers explicitly, one can use the cocartesian squares (3.28). In effect,we have the functor j : ∆ ∼ = ([1] \ Λ) o → Λ o ∼ = Λ, and the pullback functors ω ∗ n for the functors ω n , n ≥ ω ∗ : j ∗ Λ C → θ ∗ C ⊂ ρ o ∗ ♭ C ∼ = P ( C ) [1] , where θ is the functor (3.12) and P ( C ) is the path 2-category of (6.27). Inparticular, for any c ∈ C [0] , we have the natural embedding(9.2) i c : C ( c, c ) → Λ C [1] ⊂ j ∗ Λ C ⊂ Λ C sending f ∈ C ( c, c ) to the corresponding path of length 1 from c to itself. If C has one object, then (9.1) is an equivalence (but even in this good case, welose the full cyclic structure on Λ C ). If C = ∆ o I is the simplicial replacementof a category I , we will simplify notation by writing Λ I = Λ∆ o I . Example 9.2. Consider the category ∆ < with the unital monoidal struc-ture B ∆ < = ar ± (∆) o of Example 6.22. Then ∆ <♯ = Λ B ∆ < is the fullsubcategory ar < (Λ) ⊂ ar (Λ < ) spanned by arrows whose target is in Λ ⊂ Λ < .Cyclic nerves are obviously functorial with respect to 2-functors betweenthe underlying 2-categories, in that a 2-functor γ : C → C ′ induces a functor(9.3) Λ γ : Λ C → Λ C ′ cocartesian over Λ. To extend this to lax 2-functors, one can use path 2-ca-tegories in the same way as in Subsection 6.5. Namely, for any 2-category146 , the 2-functor P ( C ) → C of (6.28) has a fully faithful right-adjoint η : C → P ( C ) of (6.29), and by Lemma 2.10 (ii), Λ s of (9.3) then has a fully faithfulright-adjoint Λ η over Λ. Then for any lax 2-functor γ : C → C ′ , we considerits decomposition (6.30), and we let(9.4) Λ γ = Λ P ( γ ) ◦ Λ η : Λ C → Λ C ′ . This is a functor over Λ, not necessarily cocartesian. If γ is a 2-functor, then P ( γ ) ∼ = γ ◦ s , and since Λ η is fully faithful, we have Λ s ◦ Λ η ∼ = Id , so that(9.3) and (9.4) are consistent. By construction, we also have a functorialisomorphism(9.5) ω ∗ ◦ Λ γ ∼ = θ ∗ γ ◦ ω ∗ , where ω ∗ is the embedding (9.1). This also works in families, in the followingsense. For any category I , say that a cofibration C → ∆ o × I is a familyof -categories over I if C i = C| ∆ o × i is a 2-category for any i ∈ I . For anytwo such families C , C ′ , define a 2 -functor resp. lax -functor γ : C → C ′ asa functor over ∆ o × I whose restriction γ i : C i → C ′ i is a 2-functor resp. alax 2-functors for any i ∈ I . Then for any family C , path 2-categories P ( C i )form a family P ( C ) → ∆ o × I , with the 2-functor s : P ( C ) → C and a lax2-functor η : C → P ( C ), and just as in the absolute case, a lax 2-functor γ : C → C ′ canonically factors as γ = P ( γ ) ◦ η for a 2-functor γ ′ : P ( C ) → C ′ .For any I , we can consider the diagramΛ × I r × id ←−−−− Λ q × I l × id −−−−→ ∆ o × I, and define the relative cyclic nerve Λ( C /I ) of a family of 2-categories C over I by Λ( C /I ) = ( r × id ) ∗ ( l × id ) ∗ C . With this definition, a 2-functor γ : C → C ′ induces a functor Λ( γ ) : Λ( C /I ) → Λ( C ′ /I ) over Λ × I that restricts to Λ( C i )on any Λ × i , i ∈ I , and then (9.4) extends it to lax 2-functors. Moreover,if γ is cocartesian over a map f in I , then so is Λ( γ ). Definition 9.3. A trace theory on a 2-category C with values in a category E is a functor E : Λ C → E cocartesian over Λ. A trace functor from a unitalmonoidal category C to some E is a trace theory on B C with values in E .Definition 9.3 is a generalization of [Ka3, Definition 2.8]. In the mostbasic example, an algebra object A in a unital monoidal category C cor-responds to a lax 2-functor pt → B C , and then (9.4) provides a canonicalsection A ♯ : Λ → C ♯ of the cofibration C ♯ → Λ. If we also given a trace functor E ♯ : C ♯ → E to some category E , we can define a cyclic object E ♯ A ♯ ∈ E ; this147as essentially the main construction of [Ka3]. In general, a trace theory E on a 2-category C provides a collection of functors i ∗ c E : C ( c, c ) → E , where c ∈ C [0] is an object in C , and i c is the embedding (9.2). For a monoidalcategory C , we have only one object o ∈ B C , and i ∗ o E is a simply a functorfrom C = C ♯ [1] to E ; we call it the underlying functor of the trace functor E . Remark 9.4. Definition 9.1 suggests that Λ C really should be thought ofas a cofibration over Λ o rather than Λ, and we force it to be a cofibrationover Λ by applying (3.25). We do it for consistency with earlier definitionsof the objects F ♯ A ♯ (including the original object A ♯ of [C]).If a 2-category C is bounded, then bounded trace theories on C withvalues in some E form a well-defined category denoted Tr( C , E ); in termsof (2.17), we have Tr( C , E ) = Fun ♮ (Λ C / Λ , E ). We also have the fibrationtranspose to Λ C → Λ that we denote by Λ ⊥ C → Λ o . We denote Tr ⊥ ( C , E ) =Fun ♮ (Λ ⊥ C / Λ o , E ), and we note that (2.17) provides a natural equivalenceTr( C , E ) ∼ = Tr ⊥ ( C , E ) sending E ∈ Tr( C , E ) to(9.6) E ⊥ ∼ = l ! q ∗ r ∗ E ∈ Tr ⊥ ( C , E ) , where l , r and q are as in (2.18) for the cofibration Λ C → Λ.If γ : C ′ → C is a 2-functor between 2-categories, then the functor Λ γ of (9.4) is cocartesian over Λ, the transpose functor Λ ⊥ γ : Λ ⊥ C ′ → Λ ⊥ C is cartesian over Λ, and for any trace theory E on C , the pullback Λ γ ∗ E is a trace theory on C ′ , and we have a canonical isomorphism (Λ γ ∗ E ) ⊥ ∼ =Λ ⊥ γ ∗ E . If the 2-categories C and C ′ are bounded, we obtain a pullbackfunctor Λ γ ∗ : Tr( C , E ) → Tr( C ′ , E ) for any target category E . Lemma 9.5. Assume given a bounded -category C and a functor S : C [0] → Sets with values in non-empty sets, as in Example 7.2, and let π : C [ S ] → C be the corresponding -functor (6.7) . Then for any target category E , thepullback functor Λ π ∗ : Tr( C , E ) → Tr( C [ S ] , E ) is an equivalence of categories.Proof. We need to construct the inverse equivalence. Informally, the idea isto consider the left Kan extension Λ π ! E of a trace theory E ∈ Tr( C [ S ] , E ) andprove that, while it is not necessarily cocartesian, it nevertheless becomescartesian after applying the functor l ! q ∗ r ∗ of (9.6). Unfortunately, Λ π ! E need not even exist, so we repackage the same argument slightly differently.148amely, we define a category tw (Λ C / Λ , S ) by the cartesian product(9.7) tw (Λ C , S ) π ′ −−−−→ tw ⊥ (Λ C / Λ) R y y r ◦ q Λ C [ S ] Λ π −−−−→ Λ C , where r and q are as in (9.6), we let ϕ = l ◦ π ′ : tw (Λ C , S ) → Λ ⊥ C , andwe observe that it suffices to prove that for any E ∈ Tr( C [ S ] , E ), the leftKan extension ϕ ! R ∗ E exists, is cartesian over Λ o , and the adjunction map R ∗ E → ϕ ∗ ϕ ! R ∗ E is an isomorphism. Indeed, then the base change map(2.11) and (9.6) induce an isomorphism ϕ ! R ∗ Λ π ∗ E ∼ = E ⊥ for any E inTr( C , E ), and an isomorphism Λ ⊥ π ∗ ϕ ! R ∗ E ∼ = E ⊥ for any E in Tr( C [ S ] , E ).Since ϕ is a cofibration, Kan extensions ϕ ! can be computed by theframing (2.9), that is, by (1.13) with the comma-fibers replaced by theusual fibers. Moreover, since E is a trace theory, R ∗ E is locally constantafter restriction to each of these fibers. To describe the fiber over some c ∈ Λ ⊥ C [ n ] , choose a map f : [ n ] → [1], and let ε ( f ) : [ n − → [ n ] Λ be thecorresponding embedding (3.26). Then Example 3.5 and (3.27) provide anidentification(9.8) tw (Λ C , S ) h [ n ] ,c i ∼ = ∆ o e ( E S ( ε ( f ) ∗ c )) / [ n − , where E S = E ( S ) ◦ ν : C → E Sets is as in (6.8). But since for any [ m ] ∈ ∆and [ m ]-augmented set S , any object s ∈ e ( S ) ⊂ e ( S/ [ m ]) is initial, alocally constant functor F : ∆ o e ( S/ [ m ]) → E is constant by Lemma 3.2, andcolim ∆ o e ( S/ [ m ]) F ∼ = F ( s ). Moreover, the same is true for e ( S ) ⊂ e ( S/ [ m ]),so that in particular, the natural map colim ∆ o e ( S ) F → colim ∆ o e ( S/ [ m ]) F isan isomorphism. Applying this to E S ( ε ( f ) ∗ c ) / [ n − 1] and the restriction ofthe functor R ∗ E to the fiber (9.8), we conclude that ϕ ! R ∗ E exists and iscartesian along all maps of the form [1] → [ n ], while the adjunction map R ∗ E → ϕ ∗ ϕ ! R ∗ E is an isomorphism. Since any map [ n ] → [ m ] can becomposed with a map [1] → [ m ], ϕ ! R ∗ E inverts all cartesian maps. (cid:3) Lemma 9.5 immediately implies that an equivalence (7.9) between some2-functors γ and γ induces an isomorphism Λ γ ∗ ∼ = Λ γ ∗ between the cor-responding pullback functors (just take S to be the constant functor withvalue { , } ). Explicitly, for any 2-category C , thefiber Λ C [1] of the cyclic nerve Λ C → Λ is the category of pairs h c, f i of an149bject c ∈ C [0] and an endomorphism f ∈ C ( c, c ), with the embedding (9.2)sending f to h c, f i , while the fiber Λ C [2] consists of quadruples h c, c ′ , f, f ′ i , c, c ′ ∈ C [0] , f ∈ C ( c, c ′ ), f ′ ∈ C ( c ′ , c ). For any trace theory E , we then haveisomorphisms E ( h c, c ′ , f, f ′ i ) ∼ = E ( h c, f ′ ◦ f i ) , E ( h c, c ′ , f, f ′ i ) ∼ = E ( h c ′ , f ◦ f ′ i )provided by the two maps [2] → [1] in Λ, and these provide an isomorphism(9.9) τ f,f ′ : E ( h c, f ′ ◦ f i ) ∼ = E ( h c ′ , f ◦ f ′ i ) , a sort of a categorified trace property for E . This explains our terminology.Note that if f and f ′ form an adjoint pair h f, f ′ , a, a ′ i , then (9.9) allows todefine a natural map(9.10) E ( h c, id c i ) E ( a ) −−−−→ E ( h c, f ′ ◦ f i ) ∼ = E ( h c ′ , f ◦ f ′ i ) E ( a ′ ) −−−−→ E ( h c ′ , id c ′ i . The map is obviously invariant under automorphisms of f , so that if onechecks that it is compatible with compositions, and C is bounded, then atrace theory E defines a functor(9.11) Adj( E ) : Adj( C ) → E sending c to E ( h c, id i ), and an adjoint pair h f, f ′ , a, a ′ i to the map (9.10). If a and a ′ are invertible, (9.10) is an isomorphism, so that the functor Adj( E )inverts equivalences.In principle, it is not hard to check that (9.10) is compatible with com-positions by a direct computation. However, for homotopical applications,we will give a more invariant argument based on the description of the2-category A dj ( C ) given in Proposition 7.12. We will need the followingtechnical result. Lemma 9.6. Assume given a -category C equipped with a -functor C → eq , and a -special functor E : C → C to some category E . Then the map colim i ∗ C i ∗ E → colim C E induced by the embedding i : i ∗ C → C is an isomorphism, and its sourceexists iff so does its target.Proof. Let eq ⊂ eq be the full subcategory spanned by injective maps a : [ n ] → [ n ] with non-empty [ n ] , denote C = C × eq eq , and note that theembedding i ∗ C → C factors as i ∗ C α −−−−→ C β −−−−→ C . α is fully faithful and has a left-adjoint s : C → C induced by (7.15),so that i ∗ C ⊂ C is left-admissible, and then by (1.14), it suffices to prove thatthe left Kan extension β ! β ∗ E exists, and the adjunction map β ! β ∗ E → E isan isomorphism. But the full embedding β has a framing given by the fullsubcategories β ( c ) ⊂ C /c , c ∈ C spanned by 0-special maps, and to finishthe proof, it remains to observe that for any c ∈ C [ n ] that is not in C , β ( c ) isequivalent to the category (∆ / [ n +1]) o ∼ = ∆ o [ n +1], so that β ( c ) > ∼ = (∆ o> ) n +1 .Then since E is 0-special, E | β ( c ) > is locally constant, thus constant, and thenexact by Lemma 3.2. (cid:3) Consider the relative cyclic nerve Λ( Eq / ∆). By definition, it comesequipped with a cofibration π : Λ( Eq / ∆) → Λ, so we can define a category tw (Λ , Eq ) by the cartesian square(9.12) tw (Λ , Eq ) π ′ −−−−→ tw (Λ) R y y t Λ( Eq / ∆) π −−−−→ Λ , where t : tw (Λ) → Λ is as in (1.5). Then s of (1.6) induces a cofibration ϕ = s ◦ π ′ : tw (Λ , Eq ) → Λ o , and R composed with the cofibration Λ( Eq / ∆) → ∆ gives rise to a cofibration tw (Λ , Eq ) → ∆. For any [ n ] ∈ ∆, its fiber tw (Λ , Eq ) [ n ] ∼ = tw (Λ , V ([ n ]) is the category (9.7) for C = pt and S = V ([ n ]).Both R , ϕ are cocartesian over ∆, and restrict to the eponymous functorson the fibers tw (Λ , Eq ) [ n ] ∼ = tw (Λ , V ([ n ])).Now assume given a 2-category C , and consider the adjunction 2-category A dj ( C ) of (7.56), with the transpose fibration A dj ( C ) ⊥ → ∆. Note that since A dj ( C ) → ∆ o is semidiscrete, we actually have A dj ( C ) ⊥ ∼ = A dj ( C ) o . Thenthe evaluation functor (2.23) and the embedding (7.53) induce a functor Eq × ∆ A dj ( C ) o ∼ = Eq × ∆ A dj ( C ) ⊥ η × id −−−−→ Adj × ∆ A dj ( C ) ⊥ −−−−→ C that we denote by ev , and then (9.4) provides a functor(9.13) Λ ev : Λ( Eq × ∆ A dj ( C ) ⊥ / A dj ( C ) o ) ∼ = Λ( Eq / ∆) × ∆ A dj ( C ) o → Λ C . On the other hand, (9.12) gives rise to functors(9.14) ϕ × id : tw (Λ , Eq ) × ∆ A dj ( C ) o → Λ o × A dj ( C ) o R × id : tw (Λ , Eq ) × ∆ A dj ( C ) o → Λ( Eq / ∆) × ∆ A dj ( C ) o . Say that a map f in A dj ( C ) o is antispecial if so is its image in ∆, and saythat a functor A dj ( C ) o → E to some category E is antispecial if it inverts allantispecial maps. 151 emma 9.7. Let Λ ev be as in (9.13) , and let R , ϕ be as in (9.14) . Thenfor any trace theory E : Λ C → E on C with values in some category E , theleft Kan extension a ( E ) = ( ϕ × id ) ! ( R × id ) ∗ Λ ev ∗ E : Λ × A dj ( C ) o → E exists. Moreover, for any c ∈ A dj ( C ) o , the restriction a ( E ) c = a ( E ) | Λ × c islocally constant, and for any [ n ] ∈ Λ , the restriction a ( E ) [ n ] = E | [ n ] ×A dj ( C ) o is antispecial.Proof. For any [ n ] ∈ ∆ and c ∈ A dj ( C ) o [ n ] ⊂ A dj ( C ) o , the evaluationfunctor ev induces a lax 2-functor ev c : Eq [ n ] = ∆ o e ( V ([ n ])) → C . Forany [ l ] ∈ Λ, denote by F ( n, l ) = tw (Λ , V ([ n ])) [ l ] the fiber of the cofibration ϕ : tw (Λ , V ([ n ])) → Λ o , and let E c = R ∗ Λ( ev c ) ∗ E | F ( n,l ) . As in the proof ofLemma 9.5, the fiber F ( n, l ) is explicitly given by (9.8) that reads as F ( n, l ) ∼ = ∆ o e ( V ([ n ]) × [ l − / [ l − , and we have a ( E )([ l ] × c ) = colim F ( n,l ) E c . If [ n ] = 0, then the colimit exists by Lemma 9.5, and we in fact have a ( E ) c ∼ =Λ ε ( c ) ∗ E ⊥ , where ε ( c ) : pt → C is the embedding onto c ∈ C [0] = A dj ( C ) [0] .In particular, it is locally constant. Moreover, the functor (7.53), hence also ev and Λ ev are cocartesian over antispecial maps. Thus it suffices to provethat for any [ n ] and c , and any antispecial map f : [ m ] → [ n ] in ∆, the mapcolim F ( m,l ) F ( f ) ∗ E c → colim F ( n,l ) E c induced by the embedding F ( f ) : F ( m, l ) → F ( n, l ) is an isomorphism. More-over, it obviously suffices to prove it for m = 0. But then, we can considerthe map p : V ([ n ]) → { , } sending n to 1 and the rest to 0, with the in-duced projection p : Eq [ n ] → eq , and (7.54) immediately implies by inductionthat ev c is 0-special with respect to p . Then Λ ev c is 0-special with respectto the induced projection F ( n, l ) → eq , and we are done by Lemma 9.6. (cid:3) As a corollary of Lemma 9.7, we see that any trace theory E : Λ C → E on a bounded 2-category C gives rise to an antispecial functor A dj ( E ) o : A dj ( C ) o → Tr( pt , E ) ∼ = E , c a ( E ) c . If we take the opposite functor A dj ( E ) : A dj ( C ) → E o and compose it with ι : ι ∗ A dj ( C ) → A dj ( C ), then the resulting functor A dj ( E ) ◦ ι : ι ∗ A dj ( C ) →E o is special in the sense of Definition 6.10, thus factors through Adj( C ) o ∼ =152 ( ι ∗ A dj ( C )) by Corollary 6.12, and this gives rise to the functor (9.11) thatwe set out to construct. Explicitly, we have(9.15) Adj( E ) = ξ ⊥ ! A dj ( E ) o , where for any category I , we let ξ ⊥ : ∆ I → I be the functor sending h [ n ] , i q i to i q ( n ) ∈ I , and we extend it to 2-categories by composing with thetruncation functor C ⊥ → ∆ τ ( C ). Now assume given a bounded 2-category C , and assumethat it is pointed in the sense of Subsection 8.6, with the object o ∈ C [0] ,the embedding j o : ∆ o → C and the unital monoidal category C o = C ( o, o ), B C o ∼ = j ∗ o C . Then j o has the factorization (6.6), j o : B C o → C is a fullyfaithful embedding, and we can define the reduction functor (9.16) Red = Λ j ∗ o : Tr( C , E ) → Tr( B C o , E )for an arbitrary target category E . It turns out that in many cases, one canalso define a functor going in the other direction, and in fact reconstruct atrace theory E ∈ Tr( C , E ) from its reduction Red( E ).Firstly, assume that the target category E is cocomplete. Then assumegiven a base S of the pointed 2-category hC , o i in the sense of Definition 8.32,with the corresponding coadjont pair (8.43) and the lax 2-functor Θ of (8.44),consider the category tw (Λ C , S ) of (9.7) with its projectionsΛ C [ S ] R ←−−−− tw (Λ C , S ) ϕ −−−−→ Λ ⊥ C , and define the expansion functor Exp : Fun( C ♯o , E ) → Fun(Λ ⊥ C , E ) by(9.17) Exp( E ) = ϕ ! R ∗ ΛΘ ∗ E. Moreover, let tw (Λ C , S, eq ) = tw (Λ C , S × { , } ), with the projectionsΛ C [ S ] { eq } ∼ = Λ C [ S × { , } ] R + ←−−−− tw (Λ C , S, eq ) ϕ + −−−−→ Λ ⊥ C , and let Θ + = γ ◦ γ ⋄ : C [ S ] { eq } → C , where γ and γ ⋄ are the components ofthe decompositions (6.6) and (7.32) of the coadjoint pair (8.43). Then forany E ∈ Tr( C , E ), the embeddings i , i : C [ S ] → C [ S ] { eq } induce maps i ∗ ϕ +! R ∗ + ΛΘ ∗ + E a −−−−→ ϕ +! R ∗ + ΛΘ ∗ + E a ←−−−− i ∗ ϕ +! R ∗ + ΛΘ ∗ + E. We have i ∗ Θ + ∼ = Λ j o ◦ Θ, so that i ∗ ϕ +! R ∗ + ΛΘ ∗ + E ∼ = Exp(Red( E )), and i ∗ Θ + is the projection π : C [ S ] → C , so that i ∗ ϕ +! R ∗ + ΛΘ ∗ + E ∼ = E ⊥ by Lemma 9.5.153oreover, the fibers (9.8) of the map ϕ + carry natural projections to eq induced by C [ S ] { eq } → eq , and then Lemma 9.6 shows that the map a isan isomorphism. Altogether, we obtain a functorial map(9.18) Exp(Red( E )) → E ⊥ for any E ∈ Tr( C , E ). Note that since by definition, S ( o ) consists of a singlemap id : o → id , we also have a functorial isomorphism(9.19) Red(Exp( E )) ∼ = E ⊥ for any E ∈ Tr( B C o , E ) induced by (9.6) and (2.11). Lemma 9.8. Assume that the base S is ind-admissible in the sense of Defi-nition 8.32. Then the expansion Exp( E ) of any trace theory E ∈ Tr( B C o , E ) lies in Tr ⊥ ( C , E ) ⊂ Fun(Λ ⊥ C , E ) .Proof. Denote by F ([ n ] , c ) the fiber of the cofibration ϕ over an object h [ n ] , c i ∈ Λ ⊥ C [ n ] , and let E ([ n ] , c ) be the restriction of the functor R ∗ ΛΘ ∗ E to this fiber. As in the proof of Lemma 9.5, it suffices to prove that Exp( E )is cartesian along any map f : [1] → [ n ], [ n ] ∈ Λ. This amounts to checkingthat the mapcolim F ([1] ,c ) E ([1] , f ∗ c ) ∼ = colim F ([1] ,c ) F ∗ E ([ n ] , c ) → colim F ([ n ] ,c ) E ([ n ] , c )induced by the embedding F : F ([1] , f ∗ c ) → F ([ n ] , c ) is an isomorphism. Fixa map h : [ n ] → [1] to obtain an identification (9.8) of the fiber F ([ n ] , c ),with the corresponding identification of the fiber F ([1] , f ∗ c ) induced by thecomposition h ◦ f : [1] → [1], and let S = E S ( ε ( h ) ∗ c ) ∈ Sets / [ n − 1] and l = f (0) ∈ [ n − F factors as F ([1] , f ∗ c ) α −−−−→ F ([ n ] , c ) f −−−−→ F ([ n ] , c ) , where F ([ n ] , c ) f = ∆ o e ( S/ [ n − l ⊂ F ([ n ] , c ) = ∆ o e ( S/ [ n − α is a left-admissible embedding, so that it suffices to prove that theadjunction map β ∆ o ! β ∗ E ([ n ] , c ) → E ([ n ] , c ) is an isomorphism.If n = 1, F ([ n ] , c ) f = F ([ n ] , c ), and there is nothing to prove. If not, wecan use the framing (8.20) for β ; we then have to prove that for any object h [ m − , e q i ∈ F ([ n ] , c ) \ F ([ n ] , c ) f , with the corresponding augmented functor ε = ε ( h [ m − , e q i ) of (8.20), the augmented functor ε ∗ E ([ n ] , c ) is exact.To do this, let p : S → [ n − 1] be the augmentation map, and denote by q : [ m ] → [ n ] the map in Λ corresponding to p ◦ s : [ m − → [ n − 1] under1543.27). Choose a map g : [1] → [ m ], and denote by ω m : [ m ] → [ m ] Λ resp. ω n : [ n ] → [ n ] Λ the functors (3.28) corresponding to the maps g : [1] → [ m ]resp. q ◦ g : [1] → [ n ]. Then the 2-functor Y ( S, Θ) of Proposition 8.34sends ω ∗ n c ∈ C [ n ] to an iterated ind-polycylinder h S q / [ n ] , A i in M or ∗ ( C o ),the augmented functor (8.20) naturally lifts to an embedding ε δ : ∆ o> e ( S l ) → θ ∗ ∆ o e ( S q / [ n ]) , ε δ ( o ) = h [ m ] , ω ∗ m s i , where θ is the functor (3.12), and the lax 2-functor A restricts to a functor A δ : θ ∗ ∆ o e ( S q / [ n ]) → θ ∗ B C o ∼ = j ∗ C ♯o → C ♯o , where we identify θ ∗ B C o ∼ = j ∗ C ♯o by (9.1). Furthermore, we have Θ ∼ = Θ triv ◦ Y ( S, Θ), and by virtue of (9.5) and (8.46), we then have an isomorphism ε ∗ E ([ n ] , c ) ∼ = ε ∗ δ A ∗ δ E , so it suffices to prove that ε ∗ δ A ∗ δ E is exact. Since E isa trace theory, we have ε ∗ δ A ∗ δ ζ ([ m ]) ∗ E ∼ = ε ∗ δ A ∗ δ E , where ζ ([ m ]) is the functor(2.1) for the cofibration B C o . But since A is an iterated ind-polycylinder,then as we saw in the proof of Lemma 8.22 (iii), ζ ([ m ]) ◦ A δ ◦ ε δ is a filteredcolimit of contractible augmented functors. Since E commutes with filteredcolimits, the same then holds for ε ∗ δ A ∗ δ ζ ([ m ]) ∗ E , so that it is exact. (cid:3) We can now state and prove our reconstructiontheorem. Recall that for any 2-category C and object c ∈ C [0] , we have theembedding (9.2). Definition 9.9. For any two objects c, c ′ ∈ C [0] in a pointed 2-category hC , o i equipped with a base S , and any contractible simplicial set X , a func-tor g : ∆ o> X → C ( c, c ′ ) is S -contractible if for any s ∈ S ( c ), the composition∆ o> X g −−−−→ C ( c, c ′ ) −◦ i s −−−−→ C ( o, c ′ )is contractible in the sense of Definition 3.1. A trace theory E on the 2-category C is S -exact if for any c ∈ C [0] , contractible simplicial set X , and S -contractible functor g : ∆ o> X → C ( c, c ), the functor g ∗ i ∗ c E is exact.We note that by definition, the embedding (9.2) factors through thefiber (Λ C ) [1] ⊂ Λ C over [1] ∈ Λ, and we have (Λ C ) [1] ∼ = (Λ ⊥ C ) [1] , so (9.2) alsodefines an embedding(9.20) i ⊥ c : C ( c, c ) → Λ ⊥ C . Then for any E ∈ Tr( C , E ), we have i ∗ c E ∼ = i ⊥∗ c E ⊥ , so that E is S -exact ifand only if g ∗ i ⊥∗ c E ⊥ is exact for any S -contractible g : ∆ o> X → C ( c, c ).155 heorem 9.10. Assume given a bounded pointed -category hC , o i that ad-mits an ind-admissible base S in the sense of Definition 8.32. Then for anycocomplete target category E , the reduction functor Red of (9.16) has a fullyfaithful left-adjoint functor (9.21) Exp : Tr( B C o , E ) → Tr( C , E ) whose essential image consists of trace theories that are S -exact in the senseof Definition 9.9.Proof. By Lemma 9.8, sending E ∈ Tr( B C , E ) to its expansion Exp( E ) of(9.17) defines a functor (9.21), and we have the isomorphism (9.19) and thefunctorial map (9.18). To prove the theorem, it then suffices to shows thatfirstly, for any E ∈ Tr( B C o , E ), the trace theory Exp( E ) is S -exact, andsecondly, for any S -exact trace theory E ∈ Tr( C , E ), the map (9.18) is anisomorphism. For the first claim, take some c ∈ C [0] , and define a category tw (Λ C , S ) c by the cartesian product tw (Λ C , S ) c ϕ ′ −−−−→ C ( c, c ) i ′ c y y i ⊥ c tw (Λ C , S ) ϕ −−−−→ Λ ⊥ C , where i ⊥ c is the embedding (9.20). Then (9.8) provides an identification tw (Λ C , S ) c ∼ = ∆ o e ( S ( c )) × C ( c, c ), with ϕ ′ given by the projection to thesecond factor, and we have a projectionΘ( c ) = ΛΘ ◦ R ◦ i ′ c : ∆ o e ( S ( c )) × C ( c, c ) ∼ = tw (Λ C , S ) c → C ♯o such that i ⊥∗ c Exp( E ) ∼ = ϕ ′ ! Θ( c ) ∗ E for any E ∈ Tr( B C o ) by virtue of the basechange isomorphism (2.11). By definition, Θ( c ) factors through j ∗ C ♯o ⊂ C ♯o ,and for any h [ n ] , s i ∈ ∆ o e ( S ( c )), its compositionΘ( c ) | h [ n ] ,s i×C ( c,c ) : C ( c, c ) → ( B C o ) [ n ] ⊂ θ ∗ B C o with the embedding (9.1) is explicitly given by (7.40). In particular, itsfirst component C ( c, c ) → C o with respect to the decomposition (6.4) sends f ∈ C ( c, c ) to i ∨ s ( n ) ◦ f ◦ i s ( o ) , and the other components do not depend on f at all. Therefore for any functor g : ∆ o> X → C ( c, c ) that is S -contractiblein the sense of Definition 9.9, the pullback ( id × g ) ∗ Θ( c ) ∗ E restricts to acontractible, hence exact functor ∆ o> X → E over any h [ n ] , s i ∈ ∆ o e ( S ( c )),and then g ∗ Exp( E ) ∼ = g ∗ ϕ ′ ! Θ( c ) ∗ E ∼ = ϕ ′ ! ( id × g ) ∗ Θ( c ) ∗ E is also exact.156or the second claim, take some S -exact trace theory E ∈ Tr( C , E ), andnote that since both E ⊥ and Exp(Red( E )) are cartesian over Λ, it sufficesto prove that (9.18) is an isomorphism over [1] ∈ Λ. Take an object c ∈ C [0] ,and define a category tw (Λ C , S, eq ) c by the cartesian diagram tw (Λ C , S, eq ) c ϕ ′ −−−−→ C ( c, c ) i + c y y i ⊥ c tw (Λ C , S, eq ) ϕ + −−−−→ Λ ⊥ C . Then again, (9.8) induces an identification tw (Λ C , S, eq ) c ∼ = ∆ o e ( S ( c )) { eq } × C ( c, c ) , and we have the projectionΘ( c ) + = ΛΘ + ◦ R + ◦ i + c : ∆ o e ( S ( c )) × C ( c, c ) → Λ C . Moreover, let σ : ∆ o> → eq be the functor sending [ n ] ∈ ∆ < to the injectivemap s : [0] → [ n +1] = κ ([ n ]). Then σ ∗ ∆ o e ( S ( c )) { eq } ∼ = S ( c ) × ∆ o> e ( S ( c )),and since S ( c ) is by assumption non-empty, we can choose an element in S ( c ) and lift σ to an embedding σ c : ∆ o> e ( S ( c )) → ∆ o e ( S ( c )) { eq } . Let θ ( c, f ) = Θ( c ) + ◦ ( σ c × j f ) : ∆ o> e ( S ( c )) → Λ C , where j f : pt → C ( c, c ) is theembedding onto some f ∈ C ( c, c ), and denote F = ( id × j f ) ∗ Θ( c ) ∗ + E . Thenby the same argument as in Lemma 9.6, the map F ( o ) = colim ∆ o> e ( S ( c )) σ ∗ c F → colim ∆ o e ( S ( c )) { eq } F induced by σ c is an isomorphism, and then (9.18) is an isomorphism at f ∈ C ( c, c ) ⊂ Λ ⊥ C if and only if the augmented functor σ ∗ c F = θ ( c, f ) ∗ E is exact. Moreover, since E is cocartesian over Λ, we have θ ( c, f ) ∗ E ∼ = θ ( c, f ) ∗ ζ ( θ ( c, f )( o )) ∗ E , where ζ ( θ ( c, f )( o )) is the functor (2.1) for the cofi-bration Λ C → Λ. Since θ ( c, f )( o ) lies in the image of the embedding (9.2), wehave ζ ( θ ( c, f )( o )) ◦ θ ( c, f ) ∼ = i c ◦ g for some functor g : ∆ o> e ( S ( c )) → C ( c, c ),and what we need to check is the exactness of g ∗ i ∗ c E . However, E is byassumption S -exact, and (7.40) immediately shows that for any s ∈ S ( c ), g ◦ i s extends to the contraction ∆ o + e ( S ( c )) s ⊃ ∆ o e ( S ( c )) of Lemma 8.7, sothat g is S -contractible. (cid:3) .5 Enriched categories. As in Theorem 9.10, assume given a monoidalcategory C o and a trace functor E ∈ Tr( B C o , E ) with values in some cocom-plete category E . By virtue of Proposition 8.34, it is actually sufficientto describe the expansion Exp( E ) in the universal case C = M or ∗ ( C o ) —indeed, it is obvious from (9.17) that the expansion commutes with the pull-back Λ( Y ) ∗ with respect to the 2-functor (8.45). One problem with this isthat the 2-category M or ∗ ( C o ) is not bounded (it has too many objects andtoo few morphisms). Therefore it is necessary to replace M or ∗ ( C o ) with asufficiently large but bounded full 2-subcategory M or ∗ ( C o ) b ⊂ M or ∗ ( C o )and enlarge it if necessary (this does not change the expansion). With thisconvention in mind, let us give a more explicit description of the trace theoryExp( E ) and the corresponding functor (9.11).To simplify notation, let C = C o . Consider the embedding j o : ∆ o → Λ, j o ([ n ]) = [ n +1] and the cofibration C ♯ = Λ B C → Λ, and note that thefunctorial map ε = ε ( f ) of (3.26) induces a functor(9.22) ε ∗ : j o ∗ C ♯ → B C cocartesian over ∆ o . Definition 9.11. A bimodule over a small C -enriched category h S, A i is afunctor M : ∆ o e ( S ) → j o ∗ C ♯ over ∆ o , cocartesian over anchor maps andequipped with an isomorphism ε ∗ ◦ M ∼ = A .For any cocomplete category E , an E -valued trace functor E ∈ Tr( B C , E )restricts to a functor j o ∗ E : j o ∗ C ♯ → E , and for any bimodule M over a small C -enriched category h S, A i , we can consider the object(9.23) E ( M/A ) ♯ = π ! M ∗ j o ∗ E ∈ Fun(∆ o , E ) , where π : ∆ o e ( S ) → ∆ o is the structural cofibration. Let us then define the E -twisted trace of M by(9.24) Tr EA ( M ) = colim ∆ o E ( M/A ) ♯ ∼ = colim ∆ o e ( S ) M ∗ j o ∗ E. This is obviously functorial with respect to M , so we obtain a functor fromthe category A -bimod of h S, A i -bimodules to E .Now, by definition, objects c ∈ Λ M or ∗ ( C ) [1] are represented by 2-functors from [1] Λ to M or ∗ ( C ), and by Lemma 8.15, such a 2-functor definesa C -enrichment h S ( c ) , A ( c ) i of the 2-category ∆ o [1] Λ . This consists of a set S = S ( c ) and a lax 2-functor A ( c ) : ∆ o [1] Λ [ S ] → B C . Then by (9.4), the2-functor A ( c ) defines a functor Λ( A ( c )) : Λ[1] Λ [ S ] → C ♯ = Λ B C . The cyclic158erve Λ[1] Λ is rather large. However, since Λ was defined as a subcategoryin Cat, we have the Yoneda embedding Y : ∆ o ∼ = (Λ / [1]) o → Λ[1] Λ , and acommutative diagram(9.25) ∆ o e ( S ( c )) −−−−→ Λ[1] Λ [ S ] Λ( A ( c )) −−−−−→ C ♯o y y y ∆ o Y −−−−→ Λ[1] Λ −−−−→ Λ . The composition of the two bottom arrows is the embedding j o : ∆ o → Λ,so that the composition of the top two arrows induces a functor(9.26) M ( c ) : ∆ o e ( S ( c )) → j o ∗ C ♯o . If we let A = ε ∗ ◦ M ( c ) : ∆ o e ( S ( c )) → B C , then h S, A i is a small C -enrichedcategory, and M ( c ) is an A -bimodule. Lemma 9.12. For any trace functor E ∈ Tr( B C , E ) and any object c in thefiber Λ M or ∗ ( C ) [1] , we have a natural identification (9.27) Exp( E )( c ) ∼ = Tr EA ( M ( c )) , where M ( c ) is the functor (9.26) , and the right-hand side is the E -twistedtrace of (9.24) .Proof. Combine (9.17), (9.8), (8.46), and the definition of ev . (cid:3) Remark 9.13. The notation in (9.23) is chosen for consistency with [Ka3]where we worked out to some extent the particular case of Example 8.9.Now let us turn to the functor (9.11). For any small C -enriched category h S, A i , (9.4) provides a functor Λ( A ) : Λ∆ o e ( S ) ∼ = Λ[ S ] → C ♯ , and for anytrace functor E ∈ Tr( B C , E ), we can define the object(9.28) EA ♯ = π ! Λ( A ) ∗ E ∈ Fun(Λ , E ) , where as in (9.23), π : Λ[ S ] → Λ is the structural cofibration. This isobviously functorial with respect to A and also with respect to S , so thatwe obtain a functor Cat( C ) → Fun(Λ , E ). Composing it with the projectionTw Λ : Fun(Λ , E ) → Tr( pt , E ) of Lemma 3.11 then gives a functor(9.29) CC ( E ) : Refl( C ) → Tr( pt , E ) . C ) ⊂ Cat( C ) be the dense subcategory definedby the class of functors reflexive in the sense of Definition 8.25. We thenhave the functor(9.30) CC ′ ( E ) : Refl( C ) → Tr( pt , E )obtained by composing the natural functor Refl( C ) → Adj( M or ∗ ( C )) andthe functor Adj(Exp( E )) of (9.11). Lemma 9.14. The restriction of the functor CC ( E ) of (9.29) to the sub-category Refl( C ) ⊂ Cat( C ) is isomorphic to the functor CC ′ ( E ) of (9.30) .Proof. For any categories I and E , to construct an isomorphism E ∼ = E ′ between any two functors E, E ′ ∈ Fun( I, E ), it suffices to construct isomor-phisms E ( i ) ∼ = E ′ ( i ) and all objects i ∈ I that are compatible with all maps f : i → i ′ . Compatibility means that E ( f ) = E ′ ( f ), or more generally, thatfor any diagram (1.3), we have(9.31) E ( g ) ◦ E ′ ( f ) = E ′ ( f ) ◦ E ( g ′ ) . In our case, objects are small C -enriched categories h S, A i , and isomorphisms(9.32) Tw Λ ( EA ♯ ) = CC ( E )( h S, A i ) ∼ = CC ′ ( E )( h S, A i )are provided by the same argument as in Lemma 9.12. What we have tocheck is (9.31).By definition, morphisms f : h S , A i → h S , A i in Adj( M or ∗ ( C )) arerepresented by adjoint pairs in the 2-category M or ∗ ( C ) that correspond to C -enrichments h S ( f ) , A ( f ) i of the 2-category adj . Such an enrichment givesrise to a C -enrichment h S ( f ) , A ( f ) ◦ η i of the 2-category eq , and we canconsider the object EA ( f ) ♯ = π ! Λ( A ( f )) ∗ E ∈ Fun(Λ , E ), where we againlet π : Λ eq [ S ( f )] → Λ be natural discrete cofibration. We then have thediagram(9.33) Tw Λ ( EA ♯ ) i −−−−→ Tw Λ ( EA ( f ) ♯ ) i ←−−−− Tw Λ ( EA ♯ ) , the map i is invertible by Lemma 9.6, and in terms of (9.32), the mapAdj(Exp( E ))( f ) is given by i − ◦ i . Then if we consider f as an object inthe arrow category ar (Refl( C )), both EA ♯ and EA ♯ in (9.33) are functorialwith respect to f , and to prove (9.31), it suffices to show the same for EA ( f ) ♯ and the maps i , i . Moreover, the whole diagram (9.33) depends160unctorially on the enrichment h S ( f ) , A ( f ) i , so it suffices to check that anycommutative diagram(9.34) h S , A i h f,g i −−−−→ h S , A i h h ,r i y y h h ,r i h S ′ , A ′ i h f ′ ,g ′ i −−−−→ h S ′ , A ′ i in Cat( C ) with reflexive h f, g i and h f ′ , g ′ i gives rise to maps h : adj [ S ( f )] → adj [ S ( f ′ )] and r : A ( f ) → h ∗ A ( f ′ ) that restrict to h h , r i resp. h h , r i overthe objects 0 resp. 1 in adj .Now, for any reflexive functor h f, g i : h S , A i → h S , A i between twosmall C -enriched categories, the corresponding adjoint pair has been con-structed in Proposition 8.28. Namely, we take S ( f ) = S ⊔ S , construct thedecomposition (8.37), and consider the diagram(9.35) C (∆ o e ( σ )) w −−−−→ adj p [ S f ] δ −−−−→ adj [ S ( f )] , where w is the functor (8.38), and δ is as in (8.34). We then construct thefunctor A g , and take A w = w ∗ A g and A ( f ) = δ ∆ o ! A w . Both S ( f ), (9.35)and A g are obviously functorial in f , so that a square (9.34) gives rise to amap A g → h ∗ A ′ g , and we have a diagram w ∗ A g −−−−→ w ∗ h ∗ A g ←−−−− h ∗ w ∗ A ′ g , where the map on the right is the base change map. However, it is obviousfrom the explicit description of w ∗ given in Lemma 6.25 that the base changemap is in fact an isomorphism and can be inverted. Therefore A w is alsofunctorial in f , and then again by base change, so is A ( f ). (cid:3) Corollary 9.15. For any trace functor E , the functor (9.29) inverts equiv-alences, and identifies reflexive functors that are isomorphic as morphismsin C at ( C ) .Proof. Clear. (cid:3) Remark 9.16. In practice, one is often only interested in the functor (9.29)induced by a trace functor E , and can define it directly without goingthrough all the machinery of Theorem 9.10, Lemma 9.7 and the rest ofthe material in this section. However, Corollary 9.15 then becomes rathercumbersome to prove. 161 .6 Additional structures. Let us now describe some additional struc-tures trace theories can carry, and show that these are preserved by theexpansion functor of Theorem 9.10. Assume given a 2-functor γ : C → C ′ be-tween pointed bounded 2-categories, and assume that γ is pointed (that is, γ ( o ) = o ). Define an ind-admissible base for γ as a pair of a functor S ′ : C ′ [0] → Sets, with non-empty values, and an adjoint pair ι : adj ×C [ γ ∗ S ′ ] → C such that h γ ∗ S ′ , ι i is an ind-admissible base for C , and h S ′ , γ ◦ ι i is anind-admissible base for C ′ . Then being pointed, γ restricts to a 2-functor γ o : B C o → B C ′ o , and if we let Red, Red ′ resp. Exp, Exp ′ be the functors(9.16) resp. (9.21) for C , C ′ , we have an obvious isomorphism Λ( γ o ) ∗ ◦ Red ′ ∼ =Red ◦ Λ( γ ) ∗ that gives rise to the base change map(9.36) Exp ◦ Λ( γ o ) ∗ → Λ( γ ) ∗ ◦ Exp . A moment’s reflection shows that (9.36) is also an isomorphism: indeed, thefunctor Λ( γ ) identifies the fibers (9.8) for the cofibration ϕ in (9.17), so that(9.36) reduces to the base change isomorphism (2.11).Alternatively, one can consider the cylinder C ( γ ) of the functor γ . Thenit is a family of 2-categories over [1] in the sense of Subsection 9.1, and givingan ind-admissible base for the 2-functor γ is equivalent to giving a functor S : C ( f ) [0] → Sets, with non-empty values and cocartesian over [1], togetherwith a functor ι : adj × C ( f ) [0] [ S ] → C ( f ), cocartesian over ∆ o × I and suchthat h S, ι i restrict to an admissible base on C = C ( f ) and C ′ = C ( f ) .Now more generally, say that a family of 2-categories C → ∆ o × I overa bounded category I is pointed if it is equipped with a cocartesian section o : I → C [0] of the discrete cofibration C [0] → I . An ind-admissible base for a pointed family hC /I, o i is a pair of a functor S : C [0] → Sets, witnnon-empty values and cocartesian over I , and a functor ι : adj ×C [0] → C ,cocartesian over ∆ o × I and such that h S, ι i restricts to an ind-admissible basefor each 2-category C i , i ∈ I . We can then consider the relative cyclic nerveΛ( C /I ) → Λ × I , and say that for any category E , an E -valued trace theory of C /I is a cocartesian functor Λ( C /I ) → E . If C and I are bounded, theseform a category Tr( C , E ). The section o : I → C [0] gives rise to a familyof 2-categories B C o → ∆ o × I over I and the cocartesian full embedding j o : B C o → C , and we have the functor(9.37) Red = Λ j ∗ o : Tr( C , E ) → Tr( B C o , E ) , a relative version of (9.16). Moreover, we can consider the subcofibrationTr( C /I, E ) ⊂ Fun( C /I, E ) over I spanned by Tr( C i , E ) ⊂ Fun(Λ C i , E ), and162e have Tr( C , E ) ∼ = Sec ♮ ( I o , Tr( C /I, E )), and similarly for Tr( B C o , E ). Thefunctor (9.37) is then induced by a cocartesian functor(9.38) Red I : Tr( C /I, E ) → Tr( B C o /I, E )whose fibers are the functors (9.16) for the 2-categories C i . If E is cocom-plete, these have left-adjoint expansion functors (9.17) of Theorem 9.10, andcrucially, for any map f : i → i ′ , the adjunctions maps (9.36) for the tran-sition functor γ = f ! : C i → C i ′ are isomorphisms. Therefore Red I admits acocartesian left-adjoint(9.39) Exp I : Tr( B C o , E ) → Tr( C , E )over I o , and taking the global sections, we also obtain a left-adjoint Exp tothe functor (9.37). Next, we want to discuss the relationship betweentrace theories and symmetric monoidal structures of Subsection 4.4. Westart with a relative version of Definition 4.10 and Definition 4.14. Definition 9.17. A unital symmetric monoidal structure on a cofibration C → I is given by a cofibration B ∞ C → Γ + × I equipped with an equivalence B ∞ C| pt + × I ∼ = C such that for any i ∈ I , B ∞ C Γ + × i is a unital symmetricmonoidal structure on C i in the sense of Definition 4.10. A lax monoidalstructure on a functor γ : C → C ′ over I between two cofibrations C , C ′ /I equipped with unital symmeric monoidal structures B ∞ C , B ∞ C ′ / Γ + × I is afunctor B ∞ γ : B ∞ C → B ∞ C ′ over Γ + × I such that for any i ∈ I , B ∞ γ | Γ + × i is a lax monoidal structure on γ i : C i → C ′ i in the sense of Definition 4.14. Example 9.18. For any unital symmetric monoidal structure B ∞ C on acategory C , B ∞ C → Γ + carries a natural symmetric monoidal structure B ∞ B ∞ C = m ∗ B ∞ C , where m : Γ + × Γ + → Γ + is the smash product functor.In the situation of Example 9.18, B ∞ C induces a non-symmetric unitalmonoidal structure B C = Σ ∗ B ∞ C on C , and then B ∞ B ∞ C induces a unitalsymmetric monoidal structure B ∞ B C = ( id × Σ) ∗ B ∞ B ∞ C on B C / ∆. More-over, for any partially ordered set J and C -enriched J -augmented categories h S/J, A i , h S ′ /J, A ′ i , we can define the product A ⊠ A ′ as the composition∆ o e ( S × J S ′ ) ⊂ ∆ o e ( S/J ) × ∆ o ∆ o e ( S ′ /J ) A × A ′ −−−−→ B C × ∆ o B C → B C , where the last functor is the product on B C / ∆ o . Then the cofibrationAug( C ) / ∆ o also carries a natural unital symmetric monoidal structure, with163he unit given by section ∆ o → Aug( C ) sending [ n ] ∈ ∆ o to [ n ] × pt C , andproduct given by h S/ [ n ] , A i ⊗ h S ′ / [ n ] , A ′ i = h S × [ n ] S ′ / [ n ] , A ⊠ A ′ i . This induces unital symmetric monoidal structures on all the 2-categoriesof Proposition 8.20.Now, for any 2-category C , a unital symmetric monoidal structure B ∞ C on C / ∆ o is a family of 2-categories over Γ + , and moreover, it is pointedby the unit section Γ + → B ∞ C [0] of the commutative monoid B ∞ C [0] . Wethen have the unital symmetric monoidal structure B ∞ Λ( C ) = Λ( B ∞ C / Γ + )on its cyclic nerve Λ C / Λ, and for any category E equipped with a unitalsymmetric monoidal structure B ∞ E , we can define a multiplicative structure B ∞ E on a functor E : Λ C → E as a lax monoidal structure on the product E × π : Λ C → E × Λ, where π : Λ C → Λ is the projection. A multiplicativetrace theory is a trace theory equipped with a multiplicative structure, andif C is bounded, we denote the category of E -valued multiplicative tracetheories on C by Tr ∞ ( C , E ). The reduction functor (9.37) then induces areduction functor(9.40) Red : Tr ∞ ( C , E ) → Tr ∞ ( B C o , E ) . Moreover, (4.26) makes sense in the relative setting, so that we have a natu-ral unital symmetric monoidal structure on the category Fun( B s ∞ Λ( C ) / Γ , E ),and the full subcategory Tr( B s ∞ C / Γ , E ) ⊂ Fun( B s ∞ Λ( C ) / Γ , E ) is obviously amonoidal subcategory. Therefore (4.27) induces identifications(9.41) Tr ∞ ( C , E ) ∼ = Sec ♮ ∞ (Γ , Tr( B s ∞ C / Γ , E )) , Tr ∞ ( B C o , E ) ∼ = Sec ♮ ∞ (Γ , Tr( B s ∞ B C o / Γ , E )) , and the functor (9.40) is induced by a lax monoidal structure B ∞ Red Γ onthe functor Red Γ of (9.38). Lemma 9.19. Assume that the unital symmetric monoidal category E iscocomplete, and e ⊗ − : E → E preserves colimits for any e ∈ E . Then forany bounded symmetric monoidal -category C , the expansion functor Exp Γ of (9.39) admits a monoidal structure B ∞ Exp Γ left-adjoint over Γ + to thelax monoidal structure B ∞ Red Γ .Proof. We need to check that for any map f in Γ + , the base change mapsadjoint to the maps (2.7) for the functor B ∞ Red Γ are isomorphism. By164nduction, it suffices to consider the map m : { , } + → pt + . Moreover,it suffices to consider the universal case C = M or ∗ ( C ), and since we aredealing with trace theories, it suffices to prove that the maps are isomor-phism after evaluation at any object c ∈ Λ C [0] . Then by Lemma 9.12, thisamound to checking that the for any two sets S , S , the diagonal embed-ding ∆ o e ( S × S ) → ∆ o e ( S ) × ∆ o e ( S ) is cofinal, and this immediatelyfollows from Lemma 3.6. (cid:3) By virtue of (9.41), Lemma 9.19 immediately implies that for any tracefunctor E on C o equipped with a monoidal resp. lax monoidal structure, theexpansion Exp( E ) carries a natural monoidal resp. lax monoidal structure.Explicitly, if C = M or ∗ ( C o ), then Exp( E ) is given by (9.27) and (9.24), andfor any two C o -enriched categories h S, A i , h S ′ , A ′ i equipped with bimodules M , M ′ , we have natural maps(9.42) Tr EA ( M ) ⊗ Tr EA ′ ( M ′ ) ∼ = colim ∆ o × ∆ o E ( M/A ) ♯ ⊠ E ( M ′ /A ′ ) ♯ x colim ∆ o E ( M/A ) ♯ ⊗ E ( M ′ /A ′ ) ♯ y colim ∆ o e E ( M ⊗ M ′ /A ⊗ A ′ ) ♯ ∼ = Tr EA ⊗ A ′ ( M ⊗ M ′ )where the bottom map is induced by the monoidal structure on the tracefunctor E , and the top map is invertible by Lemma 3.6. This is the multi-plication map for the monoidal structure on Exp( E ). Remark 9.20. If one is only interested in a non-symmetric monoidal struc-ture on the categories of Proposition 8.20, then it suffices to have a non-symmetric monoidal structure on B C / ∆ o , and for this, the monoidal struc-ture on C does not have to symmetric: it suffices to ask for it to be braided.The expansion then still sends monoidal resp. lax monoidal functors tomonoidal resp. lax monoidal ones. We do not go into this for lack of in-teresting examples. Assume now given two bounded unital mono-idal categories C , C ′ . Then a monoidal functor γ : C ′ → C induces 2-functors(8.11), and by (9.36), the pullbacks Λ γ ∗ commute with expansion. If γ isonly lax monoidal, then the 2-functors (8.11) do not exists, but we still have2-functors (8.10). If we further assume that C is well-generated in the sense of165efinition 8.36, then we have the 2-functor P ◦ γ ◦ T : M or ∗ ( C ′ ) → M or ∗ ( C ).We still have the base change isomorphism(9.43) Λ γ ∗ Λ T ∗ Exp( E ) ∼ = Λ T ∗ Exp(Λ γ ∗ E )but neither Λ γ ∗ E nor Exp(Λ γ ∗ E ) are trace theories. However, assume inaddition that we have given a trace theory E ′ ∈ Tr( C ′ , E ) and a morphism α : Λ γ ∗ E → E ′ . We then have the induced morphism(9.44) Λ P ∗ Λ T ∗ Exp(Λ γ ∗ E ) → Λ P ∗ Λ T ∗ Exp( E ′ ) , and by Lemma 8.38, P ◦ T is equivalent to the identity, so that by Lemma 9.5,the target of this morphism is naturally identified with Exp( E ′ ). Combining(9.43) and (9.44), we obtain a functorial morphism(9.45) Λ P ∗ Λ γ ∗ Λ T ∗ Exp( E ) → Exp( E ′ )of E -valued trace theories on M or ∗ ( C ′ ). If C , C ′ and E are unital symmetricmonoidal, γ is lax monoidal, and E , E ′ and α are multiplicative, then (9.45)is a multiplicative map.A typical example of such a situation occurs in the following case. Con-sider the functor V : Λ → Γ ⊂ Γ + sending [ n ] ∈ Λ to the set V ([ n ] λ ) of itsvertices. Then for any bounded category C equipped with a unital symmet-ric monoidal structure B ∞ C , we have C ♯ = Λ B C ∼ = V ∗ B ∞ C . Since V factorsthrough Γ ⊂ Γ + , and pt ∈ Γ is the terminal object, (2.1) for the cofibration B ∞ C| Γ + induced then a natural functor(9.46) Id triv : C ♯ → C canonically identified with Id on C ∼ = C ♯ [1] . Thus the identity functor C → C trivially extends to a trace functor Id triv ∈ Tr( B C , C ), and more generally,any functor E : C → E to some category E lifts to a trace functor E triv = E ◦ Id triv ∈ Tr( B C , E ). Moreover, Id triv is multiplicative, so that if E issymmetric monoidal and E is multiplicative, then E triv is also multiplicative.Then if we are given another bounded unital symmetric monoidal cat-egory C ′ and a lax monoidal functor γ : C ′ → C , we can also consider thecomposition γ ∗ E : C ′ → E , and then γ induces a natural map α : γ ∗ E triv → ( γ ∗ E ) triv . Plugging it into the construction of the map (9.45), we obtain afunctorial map(9.47) Λ P ∗ Λ γ ∗ Λ T ∗ Exp( E triv ) → Exp(( γ ∗ E ) triv )of E -valued trace theories on M or ∗ ( C ′ ). Again, if E is unital symmetricmonoidal and E is lax monoidal, then (9.47) is a multiplicative map.166 .7.3 Edgewise subdivision. The last additional structure on trace the-ories that we will need is induced by the edgewise subdivision functors (3.29).Namely, observe that for any 2-category C , these functors fit into a commu-tative diagram(9.48) Λ C π l ( C ) ←−−−− Λ l C i l ( C ) −−−−→ Λ C y y y Λ π l ←−−−− Λ l −−−−→ Λ , where the square on the left is cartesian, and the functor i l ( C ) over some v : [ nl ] → [ n ] is given by the pullback v ∗ : Fun ([ n ] Λ , C ) → Fun ([ n ] Λ , C ).Fix a target category E . Definition 9.21. An h F, l i -structure on an E -valued trace theory E on C isa map F : π l ( C ) ∗ E → i l ( C ) ∗ E. If E has finite limits, then since π l is a bifibration wth finite fibers, sois π l ( C ), and there exists the right Kan extension π l ( C ) ∗ i l ( C ) ∗ E . Moreover,by (2.11), this is again a trace theory on C , and by adjunction, giving an h F, l i -structure on E is equivalent to giving a map(9.49) F † : E → π l ( C ) ∗ i l ( C ) ∗ E. Definition 9.22. The trace theory π l ( C ) ∗ i l ( C ) ∗ E is called the l -th edgewisesubdivision of the trace theory E .If C / ∆ o and E are symmetric monoidal, then Λ l C / Λ l is also symmetricmonoidal by pullback, so it makes sense to say that an h F, l i -structure F ona multiplicative trace theory E is multiplicative. If E has finite limits, thenthe l -th edgewise subdivision π l ( C ) ∗ i l ( C ) ∗ E of a multiplicative trace theory E is multiplicative, and for any multiplicative h F, l i -structure F on E , thecorresponding map (9.49) is multiplicative as well.Finally, for any functor S : C [0] → Sets, the diagram (9.48) obviouslyinduces an analogous diagram for the category tw (Λ C , S ), so that for anymonoidal category C , an h F, l i -structure F on a trace functor E on C inducesan h F, l i -structure Exp( F ) on its expansion Exp( E ). If F is multiplicative,then so is Exp( F ). We note that if C is symmetric monoidal, then givingan h F, l i -structure on E triv ∈ Tr( B C , E ) for some functor E : C → E isequivalent to giving a map(9.50) E → δ ∗ l m ∗ l E, m l : C l → C is the l -fold product, and δ l : C → C l is the diagonalembedding. Explicitly, this amounts to giving a map E ( c ) → E ( c ⊗ l ) for any c ∈ C , functorially with respect to c . Remark 9.23. Since limits do not commute with colimits, edgewise sub-division of Definition 9.22 in general does not commute with the expansionfunctor (9.17). 10 Homotopy trace theories. Recall that for any bounded category C , we have thecategory Ho( C ) of functors from C to Top + localized with respect to point-wise weak equivalences, as in Subsection 4.1, and for any commutative ring k , we also have the category Ho( C , k ) of functors from C to ∆ o k -mod, againlocalized with respect to pointwise weak equivalences, as in Subsection 4.6.For any class v of maps in C , we have the full subcategory Ho v ( C ) ⊂ Ho( C )spanned by objects E such that H o ( E ) : C → Ho inverts maps in v , andsimilarly for Ho( − , k ). For any cofibration C → I of bounded categories,say that an object E in Ho( C ) or Ho( C, k ) is homotopy cocartesian over I if so is H o ( E ) : C → Ho, and similarly, for any fibration C → I , saythat E is homotopy cartesian over I if so is H o ( E ). Then Ho ♮ ( C ) ⊂ Ho( C )resp. Ho ‡ ( C ) ⊂ Ho( C ) is the full subcategory spanned by homotopy cocarte-sian resp. homotopy cartesian functors, and similarly for Ho( − , k ). Assumegiven a cofibration C → I of bounded categories, with the transpose fibration C ⊥ → I o , and consider the corresponding diagram (2.18). Lemma 10.1. In the assumptions above, the functor (10.1) l ! ◦ q ∗ ◦ r ∗ : Ho( C ) → Ho( C ⊥ ) induces an equivalence Ho ♮ ( C ) ∼ = Ho ‡ ( C ⊥ ) , and similarly for Ho( − , k ) .Proof. Denote ϕ = r ◦ q : tw ⊥ ( C /I ) → C . Note that for any i ∈ I and c ∈ C ⊥ i ⊂ C ⊥ , the fiber tw ⊥ ( C /I ) c of the cofibration l : tw ⊥ ( C /I ) → C ⊥ is naturally identified with the right comma-category i \ I . In particular,it is homotopy contractible, with the initial object i , so that if we denote ⋄ = l ∗ ( ‡ ), the functors l ! and l ∗ induce an equivalence Ho ⋄ ( tw ⊥ ( C /I )) ∼ =Ho ‡ ( C ⊥ ). Moreover, ϕ ∗ ( ♮ ) ⊂ ⋄ , so that ϕ ∗ induces a functor(10.2) Ho ♮ ( C ) → Ho ⋄ ( tw ⊥ ( C /I )) ∼ = Ho ‡ ( C ⊥ ) , E E ⊥ = l ! ϕ ∗ E. H o ( E ⊥ )( i ) ∼ = H o ( E )( i ) for any i ∈ I , and H o ( E ⊥ ) ∼ = H o ( E ) ⊥ . Togo in the other direction, let C ′ = C o ⊥ → I be the cofibration transpose to C o → I o , consider the diagram (2.18) for the cofibration C ′ → I , and passto the opposite functors to obtain the diagram(10.3) C ⊥ ϕ o ←−−−− tw ⊥ ( C ′ /I ) o l o −−−−→ C . Then again, the fibers of the fibration l o : tw ⊥ ( C ′ /I ) → C are homotopycontractible, and ϕ o ∗ induces a functorHo ‡ ( C ⊥ ) → Ho ♮ ( C ) , E ′ E ′⊥ = l o ∗ ϕ o ∗ E ′ such that H o ( E ⊥ ) ∼ = H o ( E ) ⊥ . To finish the proof, it remains to constructfunctorial isomorphisms E ∼ = ( E ⊥ ) ⊥ and E ′ ∼ = ( E ′⊥ ) ⊥ for any E ∈ Ho ♮ ( C )and E ′ ∈ Ho ‡ ( C ⊥ ).To do this, consider the product tw ( C , C ′ ) = tw ⊥ ( C /I ) × C ⊥ tw ⊥ ( C ′ /I ).Then objects in tw ( C , C ′ ) are composable pairs c → c ′ → c ′′ of arrows in C cocartesian over I , and sending such an arrow to c → c ′′ defines a functor µ : tw ( C , C ′ ) → A ( C ), where A ( C ) ⊂ ar ( C ) is spanned by cocartesian arrows.We also have the projections ϕ, l : tw ( C , C ′ ) → C sending the pair to c resp. c ′′ , l ∗ induces an equivalence Ho ♮ ( C ) ∼ = Ho ⋄ ( tw ( C , C ′ )), where we let ⋄ = l ∗ ♮ ,and ϕ ∗ then induces a functorHo ♮ ( C ) → Ho ⋄ ( tw ( C , C ′ )) ∼ = Ho ♮ ( C ) , E ( E ⊥ ) ⊥ . Thus to identify E ∼ = ( E ⊥ ) ⊥ , it suffices to construct a functorial isomorphism ϕ ∗ E ∼ = l ∗ E for any E ∈ Ho ♮ ( C ). But we have ϕ = s ◦ µ , l = t ◦ µ , where s, t : A ( C ) → C send c → c ′ to c resp. c ′ , and we obviously have s ∗ E ∼ = t ∗ E forany E ∈ Ho ♮ ( C ). This provides the required isomorphism E ∼ = ( E ⊥ ) ⊥ . Theargument for E ′ is dual, and then the argument for the categories Ho( − , k )is identically the same. (cid:3) Now assume given a bounded 2-category C . Then its cyclic nerve Λ C is also bounded, so we may consider the homotopy categories Ho(Λ C ) andHo(Λ C , k ), k a commutative ring. Recall that we have the functor (4.1), andsimilar functors for Ho(Λ C , k ). Definition 10.2. A homotopy trace theory resp. a k -valued homotopy tracetheory on a bounded 2-category C is an object E in Ho(Λ C ) resp. Ho(Λ C , k )homotopy cocartesian over Λ. A homotopy trace functor on a unital monoi-dal category C is a homotopy trace theory on B C .169quvalently, E is a homotopy trace theory iff H o ( E ) is a trace theoryin the sense of Definition 9.3. We denote by Ho tr ( C ) = Ho ♮ (Λ C ) ⊂ Ho( λ C ),Ho tr ( C , k ) = Ho ♮ (Λ C , k ) ⊂ Ho(Λ C , k ) the full subcategories spanned by ho-motopy trace theories in the sense of Definition 10.2.Now, the reader is invited to observe that with this definition, and withLemma 10.1, all the material of Section 9 extends to homotopy trace the-ories, with identical proofs: all one has to do is to replace colimits withhomotopy colimits, Kan extensions with homotopy Kan extensions, andreinterpret E ⊥ for a homotopy cocartesian functor E in terms of (10.2).In particular, for any pointed bounded 2-category hC , o i equipped with anadmissible base S , we have a pair of adjoint functors(10.4) Red ho : Ho tr ( C ) → Ho tr ( B C o ) , Exp ho : Ho tr ( B C o ) → Ho tr ( C )given by the homotopy counterparts of (9.16) and (9.17), and similar forHo tr ( − , k ). We also have the notion of an S -homotopy exact homotopytrace theory obtained by replacing “exact” in Definition 9.9 with “homotopyexact”, and the essential image of the full expansion functor Exp of (10.4)consists of homotopy exact homotopy trace theories.All the additional structures of Subsection 9.6 also have obvious homo-topy counterparts. In particular, for monoidal structures, observe that the fi-bration Tr( B s ∞ C / Γ , Top + ) → Γ of (9.41) induces a fibration H o ( B s ∞ C / Γ) → Γ with fibers Ho( C × · · ·× C ) that is also a cofibration and carries a symmet-ric monoidal structure, and define a multiplicative homotopy trace theoryon C as an object in Sec ♮ ∞ (Γ , H o ( B s ∞ C / Γ)). Note that Lemma 9.19 thenholds with the same proof.Moreover, as in Lemma 9.7, a homotopy trace theory E on a bounded2-category C gives rise to an object a ( E ) in Ho( A dj ( C ) × Λ) that induces afunctor(10.5) Adj( E ) : Adj( C ) → Ho tr ( pt ) = Ho ∀ (Λ) , and similarly for Ho( − , k ). However, since the category Λ is not homotopycontractible, it is no longer true that Ho tr ( pt ) is equivalent to Ho.In effect, the category ∆ o is homotopy contractible, so that we at leasthave Ho ∀ (∆ o ) ∼ = Ho, and the embedding j o : ∆ o → Λ provides a pullbackfunctor j o ∗ : Ho ∀ (Λ) → Ho ∀ (∆ o ). Moreover, for any small category I , wecan define a homotopy version Tw I of the twist functor (2.13) by replacingKan extensions with homotopy Kan extensions, and then Lemma 3.11 holdswith the same proof, so that Tw Λ sends the whole Ho(Λ) into Ho ∀ (Λ o ). Ifwe now let Av Λ = Tw Λ o ◦ Tw Λ , then the same argument as in Lemma 10.1170rovides a functorial map Id → Av Λ whose evaluation E → Av Λ ( E ) at some E ∈ Ho(Λ) is an isomorphism whenever E is locally constant. Thereforeour averaging functor Av Λ : Ho(Λ) → Ho ∀ (Λ) is left-adjoint to the fullembedding Ho ∀ (Λ) ⊂ Ho(Λ). Explicitly, for any E ∈ Ho(Λ), we have(10.6) j o ∗ Av Λ ( E ) ∼ = hocolim ∆ o j o ∗ E ∈ Ho ∀ (∆ o ) ∼ = Ho , and the pullback j o ∗ then has a left-adjoint given by(10.7) Av Λ ◦ j o ! : Ho ∀ (∆ o ) → Ho ∀ (Λ) . By adjunction, the composition K = Av Λ ◦ j o ! ◦ j o ∗ is a comonad on Ho ∀ (Λ),and the composition K † = j o ∗ ◦ Av Λ ◦ j o ! is a monad on Ho ∀ (∆ o ) ∼ = Ho.To describe the comonad K , note that since homotopy colimits over ∆ o preserve finite products, the functor Tw Λ , hence also Av Λ is monoidal withrespect to pointwise smash-product, and by the projection formula, for any E ∈ Ho ∀ (Λ), we have j o ! j o ∗ E ∼ = E ∧ j o ! pt + , where pt + : ∆ o → Sets + ⊂ Top + is the constant functor with value pt + . Therefore if we let K = Av Λ ( j o ! pt + ),then K is a coalgebra object in Ho ∀ (Λ), and we have K ( E ) = E ∧ K . Wealso note that for any E ∈ Ho(Λ), we have functorial isomorphisms(10.8) hocolim Λ E ∧ K ∼ = hocolim Λ Av Λ ( E ∧ K ) ∼ = hocolim Λ K (Av Λ ( E )) ∼ = ∼ = hocolim ∆ o j o ∗ Av Λ ( E ) ∼ = hocolim ∆ o j o ∗ E, where the last isomorphism is (10.6). Dually, K † = j o ∗ K is an algebra objectin Ho ∀ (∆ o ), and we have K † ( E ) = E ∧ K † for any E ∈ Ho ∀ (∆ o ).Now, up to an equivalence, j o : ∆ o ∼ = [1] \ Λ → Λ is a discrete cofi-bration, and j o ! pt + is easy to compute; in particular, j o ∗ j o ! pt + ∈ Ho(∆ o )is canonically identified with Σ + , where Σ is the standard simplicial circleΣ : ∆ o → Sets + of (3.16). Then K † ∼ = S in Ho ∀ (∆ o ) ∼ = Ho, with thealgebra structure induced by the group structure on the circle S , and forany E ∈ Ho ∀ (Λ), the pullback j o ∗ E comes equipped with an action map(10.9) S ∧ j o ∗ E → j o ∗ E. While j o ∗ E ∈ Ho ∀ (∆ o ) ∼ = Ho is simply a homotopy type, the action (10.9)might well be non-trivial — for example, if E = K , then (10.9) is the mul-tiplication in the algebra K † .The same procedure works for k -valued homotopy trace theories, butin this case, one can say more. Namely, if k = Z , K ∈ Ho ∀ (Λ , Z ) can berepresented by an explicit complex K q in Fun(Λ , Z ) that fits into a four-termexact sequence(10.10) 0 −−−−→ Z −−−−→ K −−−−→ K −−−−→ Z −−−−→ , Z stands for the constant functor with value Z (see e.g. [Ka11, Lemma1.6]). For any k and E ∈ Ho ∀ (Λ , k ), we then have K ( E ) ∼ = E ⊗ Z K q . Thealgebra K † ( k ) = j o ∗ K q ⊗ Z k is then a DG algebra over k quasiisomorphic tothe homological chain complex C q ( S , k ) of the circle equipped with the Pon-tryagin product. This DG algebra is formal — that is, we have K † ( k ) ∼ = k [ B ],the free graded-commutative algebra in one generator B of homological de-gree 1 with trivial differential — and (10.9) can be refined to an equivalenceof categories(10.11) D ∀ (Λ , k ) ∼ = D ( k [ B ]) . Explicitly, for any E ∈ Fun(Λ , k ), the pullback j o ∗ Av Λ ( E ) is identified withthe homology object C q (∆ o , j o ∗ E ) ∈ D ( k ) by (10.6), this can be computedby the standard complex C q ( j o ∗ E ) of the simplicial k -module j o ∗ E , andthe generator B in (10.11) then acts on C q ( j o ∗ E ) by the Connes-Tsygandifferential (also known as Rinehart differential, see [L, Chapter 2] and bib-liographical comments therein). Let us now discuss homotopy trace theories from thepoint of view of the stabilization formalism of Section 4. Say that a 2-category C is half-additive iffor any c, c ′ ∈ C [0] , the category C ( c, c ′ ) is half-additive – that is, pointed withfinite coproducts – and for any f ∈ C ( c, c ′ ), all the composition functors f ◦− and − ◦ f preserve finite coproducts. In particular, C ( c, c ) is half-additivefor any c ∈ C [0] . Definition 10.3. A homotopy trace theory or a k -valued homotopy tracetheory E on a half-additive bounded 2-category C is stable if for any c ∈ C ,its restriction i ∗ c E with respect to (9.2) is stable.In particular, for any homotopy trace theory E on C , object c ∈ C [0] ,and its endomorphism f ∈ C ( c, c ), we have the object H o ( E )( h c, f i ) ∈ Hoinduced by (4.1), and if E is stable, we also have the stable Γ-space(10.12) H o st ( E )( h c, f i ) ∈ Ho st (Γ + )induced by (4.13). If f = id is the identity endomorphism, then we havethe embedding ε ( c ) : pt → C onto c , and if we let E ( c ) = Λ ε ( c ) ∗ E inHo tr ( pt ) = Ho ∀ (Λ), then H o ( E )( h c, id i ) ∈ Ho corresponds to j o ∗ E ( c ) underthe equivalence Ho ∀ (∆ o ) ∼ = Ho. To obtain a similar interpretation of the172table Γ-space (10.12), note that if we equip Γ + with the monoidal structuregiven by smash product, then B Γ + is half-additive, and for any half-additive2-category C , we have a 2-functor(10.13) m : B Γ + × C → C that induces the functors (4.12) for the half-additive categories C ( c, c ′ ). Wethen have a 2-functor ε ( c ) + = m ◦ ( id × ε ( c )) : B Γ + → C , and we can considerthe object(10.14) E ( c ) + = Λ ε ( c ) ∗ + E ∈ Ho sttr (Γ + ) . By definition, this is a stable homotopy trace functor on Γ + but it turns outthat this is the same thing as a locally constant family of stable Γ-spacesover Λ. Namely, since Γ + is symmetric monoidal, it carries the tautologicalΓ + -valued trace functor (9.46), and taking its product with the projectionΓ ♯ + → Λ provides a functor(10.15) m ♯ : Γ ♯ + → Γ + × Λ . Let Ho st♮ (Γ + × Λ) ⊂ Ho st (Γ + × Λ) be the full subcategory spanned by objectscocartesian with respect to the projection Γ + × Λ → Λ. Then (10.15) iscocartesian over Λ, so that it induces a functor(10.16) m ∗ ♯ : Ho st♮ (Γ + × Λ) → Ho sttr (Γ + ) . Lemma 10.4. The functor (10.16) is an equivalence.Proof. By definition, the fibers of the cofibration Γ ♯ + → Λ are productsΓ n + , n ≥ 1. Let Ho st (Γ ♯ + ) ⊂ Ho(Γ ♯ + ) be the full subcategory spanned byobjects whose restriction to each of these fibers is polystable in the sense ofLemma 4.9. Then (10.15) also induces a functor(10.17) m ∗ ♯ : Ho st (Γ + × Λ) → Ho st (Γ ♯ + )that has a left-adjoint Stab ◦ m ♯ ! . On each fiber Γ n + ⊂ Γ ♯ + , (10.15) re-stricts to the functor m n of (4.17) (with I = pt ), (10.16) restricts to (4.18),and by (2.11) and Remark 4.5, Stab ◦ m ♯ ! restricts to Stab ◦ m n ! . Then byLemma 4.9, the adjunction maps for the adjoint pair m ∗ ♯ , Stab ◦ m ♯ ! are iso-morphisms, so that (10.17) is an equivalence. To prove that (10.16) is alsoan equivalence, it remains to check that m ∗ ♯ E is homotopy cocartesian over173 only if so is E ∈ Ho st (Γ + × Λ). But this is obvious: any map f in Γ + × Λcocartesian over Λ lifts to a cocartesian map in Γ ♯ + . (cid:3) By virtue of Lemma 10.4, the object E ( c ) + of (10.14) defines a structureof a stable Γ-space on E ( c ) ∈ Ho ∀ (Λ), and this can be used to refine slightlythe map (10.9) for j o ∗ E ( c ). Namely, let Ho st♮ (Γ + × ∆ o ) ⊂ Ho st (Γ + × ∆ o )be the full subcategory spanned by objects cocartesian with respect toΓ + × ∆ o → ∆ o , and note that since ∆ o is homotopy contractible, we haveHo st♮ (Γ + × ∆ o ) ∼ = Ho(Γ + ). Then j o ∗ E ( c ) + defines a structure of a stableΓ-space on j o ∗ E ( c ). For any small category I , the construction of the map(10.9) works in exactly the same way relatively over the fibers of the pro-jection I × Λ → I , so that in particular, the stable Γ-space j o ∗ E ( c ) + alsocomes equipped with such a map. By adjunction, we then have a based map(10.18) j o ∗ E ( c ) + → L ( j o ∗ E ( c )) + to the free loop space L ( j o ∗ E ( c ) + ). Moreover, we have natural embeddingsΩ( j o ∗ E ( c ) + ) , j o ∗ E ( c ) + → L ( j o ∗ E ( c ) + ) in Ho st (Γ + ) onto based resp. con-stant loops. But then we can use the product map (4.7) to combine these em-beddings into a weak equivalence Ω( j o ∗ E ( c ) + ) × j o ∗ E ( c ) + → L ( j o ∗ E ( c ) + ).The map (10.18) then splits as B × Id , where B is a map(10.19) B : j o ∗ E ( c ) + → Ω j o ∗ E ( c ) + . in Ho st (Γ + ). In the k -linear case, this is the Connes-Tsygan differential thatappears in (10.11). The construction is obviously functorial, so that for anyhalf-additive 2-category C and stable homotopy trace theory E on C , thefunctor(10.20) j o ∗ ◦ Adj( E ) : Adj( C ) → Ho st (Γ + )induced by (10.5) comes equipped with a map (10.19). Moreover, if a sta-ble homotopy trace theory E ∈ Ho sttr (Γ + ) is multiplicative, then j o ∗ E ∈ Ho st (Γ + ) is an algebra object with respect to the unital symmetric monoidalstructure (4.30), and in general, for any multiplicative stable homotopy tracetheory E on a unital symmetric monoidal half-additive 2-category C , thesame applies to the value j o ∗ Adj( E )( o ) of the functor (10.20) on the unitobject o ∈ C [0] . Remark 10.5. One can show that the equivalence of Lemma 10.4 is multi-plicative, and a multiplicative stable homotopy trace theory E ∈ Ho sttr (Γ + )actually defines a locally constant family of algebra objects in Ho st (Γ + )parametrized by Λ. In particular, the algebra structure is compatible withthe map (10.19) in a certain natural way. However, we will not need this.174 roposition 10.6. For any bounded half-additive -category C , the full sub-category Ho sttr ( C ) ⊂ Ho tr ( C ) spanned by stable homotopy trace theories isleft-admissible, with the stabilization functor Stab C : Ho tr ( C ) → Ho sttr ( C ) adjoint to the embedding Ho sttr ( C ) → Ho tr ( C ) , and for any object c ∈ C [0] , wehave i ∗ c ◦ Stab C ∼ = Stab C ( c,c ) ◦ i ∗ c . The same holds for the category Ho tr ( C , k ) for any commutative ring k . Unfortunately, the stabilization procedure used in Proposition 4.4 is notcompatible with monoidal structures, and cannot be applied immediatelyin the setting of Proposition 10.6. This is a well-known problem with awell-known solution that underlies the theory of symmetric spectra of [HSS]— roughly speaking, one needs to replace the colimit (4.9) with a colimitover a more appropriate indexing category. For symmetric spectra, this isthe category of finite sets and injective maps. For trace theories, we needsomething slightly more complicated, so we first discuss the construction inan abstract setting and exhibit its essentially 2-categorical nature. For any unital monoidal category J ,the pullback κ o ∗ BJ → ∆ o satisfies the Segal condition, so that its re-duction ( κ o ∗ BJ ) red of Remark 6.5 is a 2-category. Say that J is essen-tially discrete if the cofibration ( κ o ∗ BJ ) red → ∆ o is discrete. In this case,( κ o ∗ BJ ) red ∼ = ∆ o J red is the simplicial replacement of a category J red thatwe call the reduction of the unital monoidal category J . The composition ofthe embedding ( κ o ∗ BJ ) red → κ o ∗ BJ and the projection a o ! : κ o ∗ BJ → BJ gives a 2-functor(10.21) e : ∆ o J red ∼ = ( κ o ∗ BJ ) red → BJ. Explicitly, objects in J red are objects in J , morphisms from j to j ′ are pairs h j ′′ , a i of an object j ′′ ∈ I and an isomorphism a : j ⊗ j ′′ ∼ = j ′ , and the2-functor (10.21) sends such a morphism to j ′′ . In particular, J red onlydepends on the isomorphism groupoid J ⊂ J of the category J with theinduced monoidal structure, so that J red ∼ = J red , and the reduction J red hasan initial object o ∈ J red corresponding to the unit object 1 ∈ J . Example 10.7. The unital monoidal category ∆ < is essentially discrete,and its reduction ∆ 7→ h j, id i ! x ∼ = m ( x, j ) , x ∈ M, j ∈ J over J red . Moreover, say that M is reflexive if for any j ∈ J , the functor m ( − , j ) : M → M has a right-adjoint j ∗ : M → M . Then in this case, thecofibration M red → J red is a bifibration, and (2.1) provides a functor(10.23) ω = ζ ( o ) : M red → M sending x ∈ M = M j ⊂ M red , j ∈ J to j ∗ x ∈ M . If we have two I -modules M , M , then a morphism α : M → M of I -module induces a functor α red : M red → M red over I red . If M is also reflexive, we can consider thecomposition M × J red σ −−−−→ M red α red −−−−→ M red ω −−−−→ M that gives rise to a functor(10.24) M → Fun( J red , M )as soon as J red is bounded, so that the right-hand side is well-defined.To apply this abstract machinery to stabilization, it is convenient touse both the category Top + of pointed compactly generated topologicalspaces and the category ∆ o Sets + of pointed simplicial sets. Both are unitalsymmetric monoidal with respect to the smash product, and we have anadjoint pair(10.25) Top + sing −−−−→ ∆ o sing + real −−−−→ Top + sing and the geometric realization functor real . By Milnor Theorem, the latter is symmetric monoidal (see [Dr] for amodern proof). We also have the tautological embedding Γ + → Sets + thatinduces an embedding ε : ∆ o Γ + → ∆ o Sets + .Now, (6.52) with E = Sets + and I = ∆ o provides a morphism(10.26) Fun ⊗ (Γ + , ∆ o Sets + ) → Fun ⊠ (Γ + , ∆ o Sets + )of modules over the unital monoidal category ∆ o Γ + , and its target is equiv-alent to Fun(Γ + , ε ∗ ∆ o Sets + ), where ∆ o Sets + is considered as module overitself. Then the realization functor (10.25) induces a ∆ o Γ + -module mor-phism(10.27) Fun(Γ + , ε ∗ ∆ o Sets + ) → Fun(Γ + , ε ∗ real ∗ Top + ) , and for any unital monoidal category J equipped with a monoidal functor γ : J → ∆ o Γ + , the composition of morphisms (10.26) and (10.27) induces a J -module morphism(10.28) α ( γ ) : γ ∗ Fun ⊗ (Γ + , ∆ o Sets + ) → Fun(Γ + , γ ∗ ε ∗ real ∗ Top + ) . Moreover, Top + is reflexive as a module over itself. Therefore the target ofthe morphism (10.28) is also reflexive, and if J is essentially discrete, (10.28)gives rise to the corresponding functor (10.24). Precomposing it with sing of (10.25) then gives a functor(10.29) Sp( J, γ ) : Fun(Γ + , Top + ) → Fun( J red × Γ + , Top + ) . By construction, (10.29) is functorial with respect to pairs h J, γ i — if wehave another essentially discrete unital monoidal category J ′ and a monoidalfunctor ϕ : J ′ → J , then we have a canonical isomorphism(10.30) ϕ ∗ red Sp( J, γ ) ∼ = Sp( J ′ , ϕ ∗ γ ) , where ϕ red : J ′ red → J red is induced by ϕ . In particular, we can alwaysreplace J with its isomorphism groupoid J , and by abuse of notation, wewill denote Sp( J, γ ) = Sp( J , γ ) even when γ is only defined on J . Also byconstruction, the functor (10.29) respects homotopy equivalences, so we candefine an endofunctor Stab( J, Σ) of the category Ho(Γ + ) by setting(10.31) Stab( J, γ ) = π ! ◦ Sp( J, γ ) , where π : J red × Γ + → Γ + is the projection. Then (10.30) provides a map(10.32) Stab( J ′ , ϕ ∗ γ ) → Stab( J, γ ) . J ′ = pt , this is simply a functorial map Id → Stab( J, γ ).If we take J = ∆ < , as in Example 10.7, and let γ : ∆ < → ∆ o Γ + be theonly monoidal functor that sends [0] to the simplicial circle Σ of (3.16), thenStab(∆ < , γ ) is precisely the stabilization functor Stab of (4.9) constructed inProposition 4.4. However, Γ + is symmetric monoidal; therefore we can alsotake J = Γ, as in Example 10.8, with the monoidal functor γ : Γ → ∆ o Γ + of Example 4.19 sending V ([0]) = pt ∈ Γ to Σ. Lemma 10.9. The map Stab = Stab(∆ < , V ∗ γ ) → Stab(Γ , γ ) of (10.32) induced by the embedding V : ∆ < → Γ of Example 10.8 is an isomorphism.Proof. By (10.30), for any X ∈ Ho(Γ + ), V ∗ red Sp(Γ , γ )( X ) ∈ Ho( N × Γ + ) isrepresented by the inductive system Ω n B n X of (4.9), so that in particular,it is cocartesian over N for stable X . Since every map in Γ red is isomorphicto a map in the image of V red , Sp(Γ , γ )( X ) then is cocartesian over Γ red , andif we let j : pt → Γ red be the embedding onto the initial object o ∈ Γ red , theadjunction map Sp(Γ , γ )( X ) → ( j × id ) ! X is an isomorphism. Therefore fora stable X , the canonical map X → Stab(Γ , γ )( X ) is an isomorphism, andto finish the proof, it remains to show that Stab(Γ , γ )( X ) is stable for any X .Moreover, again by (10.30), the functor Ho(Sp(Γ , γ )( X )) : Γ red → Ho(Γ + )sends a set S ∈ Γ red of cardinality n to Ω n B n X , and the latter is n -stable.Let Γ ≥ nred ⊂ Γ red be the subcategory of sets of cardinality ≥ n , and assumefor the moment that we know the following: • for any n ≥ Y ∈ Ho(Γ red ), the natural map hocolim Γ ≥ nred Y | Γ ≥ nred → hocolim Γ red Y is an isomorphims.Then we are done: by the base change isomorphism (2.11), Stab(Γ , γ )( X )is n -stable for any n , thus stable.The argument for ( • ) is non-trivial but well-known (apparently it goesback to [I, Proposition VI.4.6.12]). Since it is also very short, we reproduce itfor the convenience of the reader. By induction on n , it suffices to show thatthe embedding Γ ≥ n +1 red ⊂ Γ ≥ nred is homotopy cofinal for any n . By Remark 1.17,we only have to consider the right comma-fiber over the single object in Γ ≥ nred that is not in Γ ≥ n +1 red , and for any n , this right comma-fiber is equivalent tothe category I = Γ ≥ red of non-empty finite sets and injective maps. To provethat it is homotopy contractible, consider the product I × I , the diagonalembedding δ : I → I × I and the coproduct functor µ : I × I → I . We then178ave a natural map Id → δ ◦ µ , and also a map Id → µ ◦ δ (say given by theembedding S → S ⊔ S onto the first copy of S ). Then by Lemma 4.1, δ is aweak equivalence. Then so are its one-sided inverses p , p : I × I → I givenby the projection onto the two factors, and then also the section i : I → I × I of the projection p given by the embedding onto I × S for some fixed S ∈ I .But then p ◦ i factors through the embedding pt → I onto S . (cid:3) We now observe that the con-struction of the functor (10.24) works in the relative setting. Namely, as inDefinition 9.17, define a unital monoidal structure on a cofibration C → I as a cofibration B C → ∆ o × I whose restriction to ∆ o × i , i ∈ I is aunital monoidal structure on C i . Moreover, define a C -module as a cofibra-tion M → ∆ o × I , with the fiber M = M| [0] × I , equipped with a functor µ : M → B C over ∆ o × I that is cocartesian over all maps f × f ′ , f special,and such that the functor ζ ([0]) × ( ρ o × id ) ∗ µ : M × I ( ρ o × id ) ∗ B C is an equivalence of categories. Then for any i ∈ I , the restriction M| ∆ o × i isa C i -module in the sense of Definition 6.30, and we have the action functor(10.33) m : M × I C → M. Moreover, if M is bounded, and we have a cocomplete category E , then thefibration Fun( M/I, E ) ⊥ → I is also a cofibration, with transition functorsgiven by left Kan extension ( f ! ) ! with respect to transition functors f ! of thecofibration M → I , and we have the action functor(10.34) m : Fun( M/I, E ) ⊥ × I C → Fun( M/I, E ) ⊥ whose fiber m i over some i ∈ I is the functor (6.47) for the C i -module M i , andthe maps (2.7) are the base change maps (2.11) induced by the correspondingmaps for the functor (10.33). Then identically the same construction as inSubsection 6.6 provides a C -module Fun ⊗ ( M /I, E ) with Fun ⊗ ( M , E ) | [0] × I ∼ =Fun( M/I, E ) ⊥ and the action functor (10.34). Moreover, say that a unitalmonoidal category J/I is essentially discrete if so is each of the fibers J i , i ∈ I ; then for such a J , we have the cofibration J red → I , and if J red isbounded, then for any morphism α : M → M of J -modules with reflexive M , we have the relative version(10.35) M → Fun( J red /I, M )179f the functor (10.24).To apply this to stabilization, take an integer n ≥ 1, let m : Γ n + → Γ + bethe iterated smash product functor, and note that by induction on n , for anyhalf-additive category E that has kernels, the composition m ∗ ◦ m † of the em-bedding m † : E → Fun(Γ + , E ) and the pullback functor m ∗ : Fun(Γ + , E ) → Fun(Γ n + , E ) is fully faithful and admits a right-adjoint. Therefore as in Sub-section 6.6, we have an embedding m ∗ E ⊗ ⊂ Fun(Γ n + , E ) of Γ n + -modules andits right-adjoint. We also have the morphisms (6.51) and (6.52) with Γ + re-placed by Γ n + . More generally, consider the cyclic nerve Γ ♯ + → Λ of the cat-egory Γ + , with the monoidal structure induced by the symmetric monoidalstructure on Γ + , and let Id triv : Γ ♯ + → Γ + be the functor (9.46). Then assoon as E is cocomplete, we have a fully faithful embedding Id ∗ triv ◦ m † : Id ∗ triv E → Fun ⊗ (Γ ♯ + / Λ , E )of Γ ♯ + -modules, and for any bounded I and half-additive cocomplete E withkernels, we have a morphism(10.36) Fun ⊗ ( I × Γ ♯ + / Λ , E ) → Id ∗ triv Fun ⊠ ( I × Γ + , E )of Fun( I, Γ ♯ + / Λ)-modules, a relative version of the morphism (6.52). Wecan now take E = Sets + and I = ∆ o , let J = Γ ♯ , where Γ is the essentiallydiscrete symmetric monoidal category of Example 10.8, equip it with thefunctor γ ♯ : Γ ♯ → Fun(∆ o , Γ ♯ + / Λ) induced by γ : Γ → ∆ o Γ + of Lemma 10.9,replace (10.24) with (10.35), and repeat the construction of Subsection 10.2.2to obtain a functor(10.37) Sp(Γ ♯ , γ ♯ ) : Fun(Γ ♯ + / Λ , Top + ) ⊥ → Fun(Γ ♯red × Λ Γ ♯ + / Λ , Top + ) ⊥ over Λ, a relative version of the functor (10.29). It still respects weak equiv-alences, thus descends to a functor H o (Γ ♯ + / Λ) → H o (Γ ♯red × Λ Γ ♯ + / Λ), andwe can further define a functor(10.38) Stab Λ = π ! ◦ Sp(Γ ♯ , γ ♯ ) : H o (Γ + / Λ) → H o (Γ + / Λ)over Λ, where π : Γ ♯red × Γ ♯ + → Γ ♯ + is the projection. Lemma 10.10. The functor (10.38) is cartesian over any map f : [ n ] → [1] .Proof. For any [ q ] ∈ Λ, we have (Γ ♯ + ) [ q ] ∼ = Γ q + , and under these identifica-tions, the transition functor f ! corresponding to the map f is isomorphic to180he iterated product functor m : Γ n + → Γ + . For any X ∈ H o (Γ ♯ + / Λ) [1] ∼ =Ho(Γ + ), we have Stab Λ ( X ) = Stab(Γ , γ )( X ), where Stab(Γ , γ ) is as inLemma 10.9. Moreover, by construction, the functor (10.37) is cocarte-sian, so that Stab Λ ( m ∗ X ) ∼ = m ∗ Stab(Γ n , ϕ ∗ γ )( X ), where ϕ : Γ n → Γ isthe iterated product functor for the monoidal structure on Γ (that is, theiterated coproduct of finite sets). What we have to check, then, is that themap (10.32) induced by ϕ is an isomorphism. But by Lemma 10.9, its targetis isomorphic to Stab, and exactly the same argument proves that the sameis true for its source. (cid:3) Proof of Proposition 10.6. Consider the product B Γ + × C , with the corre-sponding 2-functor (10.13) and the 2-functor t = ε ( o ) × id : C → B Γ + × C ,and consider the corresponding diagramΛ C Λ t −−−−→ Γ ♯ + × Λ Λ C Λ m −−−−→ Λ C , of cyclic nerves and functors cocartesian over Λ. Define an endofunctor Stabof the homotopy category Ho(Γ ♯ + × Λ Λ C ) as the compositionHo(Γ ♯ + × Λ Λ C ) Sp(Γ ♯ ,γ ♯ ) −−−−−−→ Ho(Γ ♯red × Λ Γ ♯ + × Λ Λ C ) π ! −−−−→ Ho(Γ ♯ + × Λ Λ C )where π : Γ ♯red × Λ Γ ♯ + × Λ Λ C → Γ ♯ + × Λ Λ C is the projection, and Sp(Γ ♯ , γ ♯ )is induced by the functor (10.38) for the cofibration Γ ♯ + × Λ Λ C → Λ C . LetStab C = Λ t ∗ ◦ Stab ◦ Λ m ∗ : Ho(Λ C ) → Ho(Λ( C )) . Then we have a functorial map Id → Stab C , and by definition, we have i ∗ c ◦ Stab C ∼ = Stab C ( c,c ) ◦ i ∗ c for any c ∈ C [0] . Thus to prove the claim, it sufficesto check that for any homotopy trace theory E , Stab C ( E ) is a homotopy tracetheory. But since every map [ n ] → [ m ] in Λ can be composed with a map[ m ] → [1], it suffices to check that Ho(Stab C ( E )) is cocartesian over maps[ n ] → [1], and this immediately follows from Lemma 10.10. The argumentfor Ho tr ( − , k ) is identically the same. (cid:3) For any bounded half-additive pointed 2-category hC , o i , the restriction functor (9.16) commutes with stabilization,thus induces a functor Red st : Ho sttr ( C ) → Ho sttr ( B C o ), and similarly forHo( − , k ). In the k -linear case, stabilization commutes with homotopy col-imits, hence also with expansion, so that (9.17) provides a fully faithful181eft-adjoint functor Exp st ∼ = Exp ho to Red st . In the absolute case, Red st still admits a fully faithful left-adjoint given by(10.39) Exp st = Stab C ◦ Exp ho : Ho sttr ( B C o ) → Ho sttr ( C ) , while (4.34) provides an isomorphism(10.40) ϕ ◦ Ext st ∼ = Exp st ◦ ϕ. However, the absolute Stab C no longer commutes with Exp ho , and it isnot true that the essential image of the functor (10.39) consist of stable S -homotopy exact trace theories. To describe this essential image, we need tomodify the notion of homotopy exactness. It turns out that the appropriatecondition in the stable case is actually shorter. Definition 10.11. For any bounded pointed half-additive 2-category hC , o i equipped with a base S , a stable homotopy trace theory E ∈ Ho sttr ( C ) is stably S -homotopy exact if for any object c ∈ C [0] and S -contractible functor g : ∆ o> → C ( c, c ), the pullback g ∗ i ∗ c E ∈ Ho st (∆ o> ) is homotopy exact.The difference with Definition 9.9 is that we do not consider an additionalcontractible simplicial set X . Let us show that this is enough. Lemma 10.12. In the assumptions of Definition 10.11, assume further that C is large, and the base S is ind-admissible. Then a stable homotopy tracetheory E ∈ Ho sttr ( C ) is stably S -homotopy exact if and only if the naturalmap E → Exp st (Red st ( E )) is an isomorphism.Proof. For any small category I , with the tautological projection τ : I → pt ,the embedding e : I → I > , and the embedding i : pt → I > onto o ∈ I > , say that an object E ∈ Ho st (Γ + × I > ) is stably homotopy exact if theaugmentation map Stab(( id × τ ) ! ( id × e ) ∗ E ) → ( id × i ) ∗ E is an isomorphismin Ho st (Γ + ). Since hocolim ∆ o commutes with finite products, an object E ∈ Ho st (Γ + × ∆ o ) is stably homotopy exact if and only if its restriction to pt + × ∆ o is homotopy exact. In particular, for any bounded half-additivecategory E and small category I , a functor γ : I > → E uniquely extends toa half-additive functor γ + : Γ + × I > → E , and if I = ∆ o> , then γ ∗ + E forsome E ∈ Ho st ( E ) is stably homotopy exact if and only if γ ∗ E is homotopyexact.If the bounded half-additive category E is also large, then it has filteredcolimits, thus all coproducts. Then for any simplicial set X with the pro-jection π : ∆ o> X = (∆ o X ) > → ∆ o> , and any functor g : ∆ o> X → E ,182he left Kan extension ( id × π ) ! g + = (( id × π ! g ) + : Γ + × ∆ o> → E exists. If X is contractible and g is contractible in the sense of Definition 3.1, then( id × π ) ! g is also contractible. Moreover, for any X and any E ∈ Ho st ( E ),we can consider the natural map(10.41) Stab(( id × π ) ! g ∗ + E ) → (( id × π ) ! g + ) ∗ E, and by the same argument as in the proof of Proposition 4.23, (4.15) showsthat (10.41) is an isomorphism in Ho st (Γ + × ∆ > ). Therefore ( id × g + ) ∗ E isstably homotopy exact iff (( id × π ) ! g + ) ∗ E is homotopy exact.Applying this in the setting of the Lemma, we conclude that E ∈ Ho sttr ( C )is stably S -homotopy exact if and only if for any contractible X ∈ ∆ o Sets + ,any c ∈ C [0] , and any functor g : ∆ o> X → C ( c ) that is S -contractible inthe sense of Definition 9.9, the object g ∗ + i ∗ c ∈ Ho st (Γ + × ∆ o> X ) is stablyhomotopy exact. To finish the proof, it remains to apply the same argumentas in Theorem 9.10, with Exp replaced by Exp st of (10.39). (cid:3) One can also rewrite Definition 10.11 in the following way. For anybounded 2-category C , the cofibration C ∆ = Fun(∆ o , C / ∆ o ) → ∆ o is a 2-category — its objects are the objects c ∈ C [0] , and morphism categoriesare C ∆ ( c, c ′ ) = Fun(∆ o , C ( c, c ′ )), so that morphisms in C ∆ are simplicialobjects in the morphism categories C ( c, c ′ ). We have the evaluation functor ev : ∆ o × C ∆ → C and the projection π : ∆ o × C ∆ → C ∆ , these induces thecorresponding functors on cyclic nerves, and for any homotopy trace theory E ∈ Ho tr ( C ), we can define its simplicial extension E ∆ by E ∆ = π ! ev ∗ E ∈ Ho tr ( C ∆ ) . If C is half-additive, then C ∆ is also half-additive, and if E is stable, then E ∆ is stable. Moreover, assume that C is pointed, and hC , o i is equippedwith a base S . Then any functor g : ∆ o> → C ( c, c ), c ∈ C [0] defines objects g = g | ∆ o ∈ Fun(∆ o , C ( c, c )), g ( o ) ∈ C ( c, c ), and an augmentaion map(10.42) a : g → g ( o )in Fun(∆ o , C ( c, c )) ⊂ Λ C ∆[1] ⊂ Λ C ∆ whose target is the constant simplicial ob-ject with value g ( o ). Say that an augmentation map (10.42) is S -contractible if so is the functor g : ∆ o> → C ( c, c ). Then E is stably S -homotopy exact ifand only if H o ( E ∆ ) inverts all S -contractible augmentation maps (10.42).This interpretation is especially useful if we take the unital monoidalcategory k -mod fl of flat modules over a commutative ring k , and let C = A lg ( k ) ⊂ M or ∗ ( k -mod fl ) be the full subcategory in the corresponding183orita 2-category spanned by enriched categories with one object. Thenexplicitly, objects in A lg ( k ) are flat k -algebras A , and morphisms from A to A are modules M over A o ⊗ k A that are flat over A . In particular, A lg ( k )( A, A ) ∼ = A -bimod fl ⊂ A -bimod is the category of A -bimodules thatare flat as right A -modules (“right-flat”). By virtue of the Dold-Kan equiv-alence, morphisms in A lg ( k ) ∆ are then chain complexes of such moduleconcentrated in non-negative homological degrees. The 2-category A lg ( k ) ispointed, with o being k itself, and it has a standard ind-admissible base S formed by the unit maps k → A , A ∈ A lg ( k ) [0] . Definition 10.13. A homotopy trace theory E ∈ Ho tr ( A lg ( k )) is homoto-py-invariant if for any A ∈ A lg ( k ) [0] , Ho( i ∗ A E ∆ ) : C q ( A -bimod) → Ho invertsquasiisomorphisms. Lemma 10.14. A stable homotopy trace theory E ∈ Ho sttr ( A lg ( k )) is homo-topy-invariant in the sense of Definition 10.13 if and only if it is stably S -homotopy exact with respect to the standard base S .Proof. By the Dold-Kan equivalence, an augmented simplicial object ∆ o> → A -bimod for some A ∈ A lg ( k ) [0] is the same thing as a triple h M q , M, a i of a complex M q ∈ C ≥ ( A -bimod), an object M ∈ A -bimod and a map a : M q → M , where M is treated as a complex placed in degree 0. Such atriple is S -contractible if and only if the cone M ′ q of the map a is contractibleas a complex of right A -modules (“right-contractible”). In this case, a isa quasiisomorphism, so that a homotopy-invariant stable homotopy tracetheory E on A lg ( k ) is stably S -homotopy exact.To prove the converse, note that every short exact sequence(10.43) 0 −−−−→ M f −−−−→ M ′ −−−−→ M ′′ −−−−→ A -bimod fl gives rise to a cocartesian square M −−−−→ f y y f ′ M ′ −−−−→ M ′′ that we can interpret as a functor γ : [1] = V > → A -bimod fl ⊂ Λ A lg ( k ).If (10.43) is split, then for any E ∈ Ho sttr ( A lg ( k )), γ ∗ E is stably homo-topy exact in the same sense as in the proof of Lemma 10.12. Moreover,since hocolim ∆ o commutes with stabilization, the same is true for E ∆ and a184equence (10.43) in C ≥ ( A -bimod fl ), and it suffices to require that the se-quence is termwise split. However, it is well-known that homotopy cocarte-sian squares in Ho st (Γ + ) coincide with homotopy cartesian ones. Thereforefor any termwise split sequence (10.43) in C ≥ ( A -bimod), H o ( E ∆ ) invertsthe map f iff it inverts the map f ′ (that is, annihilates M ′′ ).Now assume given a stable homotopy trace theory E ∈ Ho sttr ( A lg ( k ))that is stably S -homotopy exact, and note that for any map f : M q → M ′ q in C ≥ ( A -bimod fl ), with cone C q and cylinder N q , we have termwise-splitshort exact sequences0 −−−−→ M q −−−−→ N q −−−−→ C q −−−−→ , −−−−→ M ′ q −−−−→ N q −−−−→ C ′ q −−−−→ , where the cone C ′ q of the identity map id : M q → M q is contractible. There-fore H o ( E ∆ ) inverts f if and only if it annihilates C q . But if f is a quasi-isomorphism, C q is an acyclic complex of right-flat A -bimodules. Thereforeit is a filtered colimit of acyclic complexes of right-projective A -bimodules,and these are right-contractible, thus annihilated by H o ( E ∆ ). (cid:3) By virtue of Lemma 10.14 and Lemma 10.12, any homotopy trace functoron the unital monoidal category k -mod fl extends uniquely and canonicallyto a homotopy-invariant stable homotopy trace theory E on the 2-category A lg ( k ), with its simplicial extension E ∆ ∈ Ho sttr ( A lg ( k ) ∆ ). Moreover, we caninvert quasiisomorphisms in the morphism categories of A lg ( k ) ∆ to obtaina 2-category A lg ( k ) D whose objects are again flat k -algebras, and whosemorphism categories are given by A lg ( k ) D ( A , A ) = D ≤ ( A o ⊗ k A ) ⊂ D ( A o × k A ) , where D ( A o ⊗ k A ) is the derived category of left A o ⊗ k A -modules. Thenbeing homotopy-invariant, E ∆ descends to a homotopy trace theory E D on A lg ( k ) D . This is useful to know even if one is only interested in k -algebrasand bimodules since A lg ( k ) D ⊃ A lg ( k ) has more adjoint pairs. In effect, anymultiplicative, possibly non-unital map f : A → B between two k -algebrasgives rise to a morphism in A lg ( k ) given by f (1) B ∈ A o ⊗ k B -bimod fl , andthis morphism becomes reflexive in A lg ( k ) D . Thus in particular, the functor(10.5) associated to E D induces a functor(10.44) Alg( k ) ⊂ Adj( A lg ( k ) D ) → Ho tr ( pt ) , where Alg( k ) is the category of flat k -algebras and non-unital maps. Explic-itly, any morphism in A lg ( k ) can be represented by a simplicial pointwise-flat185 o ⊗ k A -module, and by Proposition 8.28, these correspond to morphisms in C at ( k -mod fl ) that are reflexive in the sense of Definition 8.25. Therefore byLemma 9.14, the functor (10.44) is the homotopical version of the functor(9.29): it sends an algebra A to the homotopical averaging Av Λ ( EA ♯ ) of theobject EA ♯ ∈ Ho(Λ).Finally, we note that Definition 10.13, Lemma 10.14 and the discussionabove have an obvious counterpart for the categories Ho( − , k ′ ), where k ′ isa commutative ring (possibly different from k ). The proofs are identicallythe same, and we leave the details to the reader. 11 Topological Hochschild Homology. Fix a commutative ring k . The shortest wayto define Hochschild homology modules HH q ( A/k, M ) of a flat k -algebra A with coefficients in an A -bimodule M is by using the functor Tr A ofExample 1.22: one considers the object(11.1) CH q ( A/k, M ) = L q Tr A ( M ) ∈ D ( k ) , and defines HH q ( A/k, M ) as its homology modules. If we have anothercommutative ring k ′ with a map f : k ′ → k , a flat k ′ -algebra A , and abimodule M over A ⊗ k ′ k , then we have a taugolocal quasiisomorphism(11.2) CH q ( A/k ′ , f ∗ M ) ∼ = CH q ( A ⊗ k ′ k/k, M ) , where as in Example 5.5, f ∗ M is M considered as an A -bimodule by re-striction of scalars. If we have two flat k -algebras A , A ′ , then Tr A ⊗ k A ′ ∼ =Tr A ◦ Tr A ′ ∼ = Tr A ′ ◦ Tr A , and for any bimodules M , M ′ flat over k , we havethe K¨unneth isomorphism(11.3) CH q ( A, M ) L ⊗ k CH q ( A ′ , M ′ ) ∼ = CH q ( A ⊗ k A ′ , M ⊗ k M ′ ) . If one uses the standard bar resolution of the bimodule M to compute thederived functor, one obtains an explicit Hochschild complex CH q ( A/k, M )representing (11.1) with terms CH i ( A/k, M ) = A ⊗ k i ⊗ k M and a certaindifferential b . If M = A is the diagonal bimodule, one shortens HH q ( A/k, A )to HH q ( A/k ) and CH q ( A/k, A ) to CH q ( A/k ). For more details, we referthe reader to [L] or [FT].More generally, if A q is a flat DG algebra over k , with a DG bimodule M q ,one defines CH q ( A q /k, M q ) by taking the Hochschild complex CH q ( − , − )186ermwise, and then taking the product-total complex of the resulting bi-complex, and one denotes by HH q ( A q /k, M q ) the homology modules of thecomplex CH q ( A q , M q ) (for details, see [Ke]). Topological Hochschild Homology T HH was introduced by B¨okstedt [B¨o]as a generalization of Hochschild homology to the absolute setting: forany associative ring spectrum A and A -bimodule spectrum M , one con-structs a simplicial spectrum CH ( A, M ), a generalization of the complex CH q ( − , − ), and defines the spectrum T HH ( A, M ) as the homotopy colimit hocolim ∆ o CH ( A, M ). To make this work, one needs to understand a “ringspectrum” in an appropriate sense, and there are various ways to do it. How-ever, in any approach, an associative algebra A over a commutative ring k gives an example of an associative ring spectrum, so it makes sense to speakof the spectrum T HH ( A, M ). Moreover, in such a situation, T HH ( A, M )is automatically a module over k , so that a posteriori , it is not a spectrumbut a complex T CH q ( A, M ) ∈ D ≤ ( k ) of k -modules. Its homotopy groups T HH q ( A, M ) = π q ( T HH ( A, M )) are then homology groups of the complex T CH q ( A, M ) (in particular, they are k -modules in a natural way).It was realized pretty soon after [B¨o] that in the particular case of analgebra A over a ring k , T HH ( A, M ) actually admits several alternativedefinitions, some of them manifestly k -linear. One such uses bifunctor ho-mology of Subsection 1.4. Say that a right module P over a flat k -algebra A is finite free if P ∼ = V ⊗ k A for a finitely generated projective k -module V , and let P ( A ) be the category of finite free left A -modules. Then P ( A )is k -linear and small, and any A -bimodule M defines a k -bilinear functor P ( M ) : P ( A ) o × P ( A ) → k -mod sending V × V ′ to Hom A ( V, V ′ ⊗ A M ).The correspondence M P ( M ) is an equivalence between A -bimod andFun k ( P ( A ) o × P ( A ) , k ) that intertwines Tr A with Tr P ( A ) , so that we have HH q ( P ( A ) /k, P ( M )) ∼ = HH q ( A/k, M ). The absolute bifunctor homologyis then naturally identified with T HH ( A, M ) – we have(11.4) CH q ( P ( A ) , P ( M )) ∼ = T CH q ( A, M ) ∈ D ≤ ( k ) ∼ = Ho st (Γ + , k ) , and the map (1.33) provides a functorial map T CH q ( A, M ) → CH q ( A, M ).The homology groups HH q ( P ( A ) , P ( M )) are thereby identified with the ho-motopy groups T HH q ( A, M ). If we have another flat k -algebra A ′ equippedwith a k -flat bimodule M ′ , then we have T CH q ( A, M ) L ⊗ k T CH q ( A ′ , M ′ ) ∼ = CH q ( P ( A ) × P ( A ′ ) , P ( M ) ⊠ k P ( M ′ )) , and the tensor product functor −⊗ k − : P ( A ) × P ( A ′ ) → P ( A ⊗ k A ′ ) induces187 K¨unneth map(11.5) T CH q ( A, M ) L ⊗ k T CH q ( A ′ , M ′ ) → T CH q ( A ⊗ k A ′ , M ⊗ k M ′ ) . Unlike (11.5), the map (11.5) is usually not an isomorphism.To make bifunctor homology more amenable to computation, one canuse the stabilization functor (4.37) and the equivalence (5.2). One observesthat the equivalence Fun k ( P ( A ) , k ) ∼ = A -mod extends to an equivalence D ( Q q ( P ( A ) /k )) ∼ = D ( Q q ( A/k )), and similarly for P ( A ) o . For bimodules, wethen obtain an equivalence(11.6) Ho st ( P ( A ) o × P ( A ) , k ) ∼ = D ≤ ( Q q ( A/k ) o ⊗ k Q q ( A/k )) , where by abuse of notation, Ho st ( P ( A ) o × P ( A ) , k ) ⊂ Ho( P ( A ) o × P ( A ) , k )is the full subcategory spanned by objects stable with respect to both P ( A )and P ( A ) o . Taking stabilization with respect to both P ( A ) o and P ( A ), inany order, and using (11.6), we obtain a functorStab : Fun( P ( A ) o × P ( A ) , k ) → D ≤ ( Q q ( A/k ) o ⊗ k Q q ( A/k )) . If we compute Stab by the resulution (4.39), then for any M ∈ Fun( P ( A ) o × P ( A ) , k ), the degree-0 term Stab( M ) of the resulting complex Stab( M ) q is M itself, and we have an obvious map Tr P ( A ) ( M ) → HH ( Q q ( A ) , Stab( M ) q )that gives rise to a map(11.7) CH q ( P ( A ) , M ) = L q Tr P ( A ) M → CH q ( Q q ( A ) , Stab( M ) q )in D ≤ ( k ). For any object P ∈ P ( A ) and functor N ∈ Fun k ( P ( A ) , k ), themap (11.7) for M = k P ⊠ N is a quasiisomorphism by (1.27) and (1.34), andsince every M ∈ Fun k ( P ( A ) o × P ( A ) , k ) admits a resolution by functors ofthis form, (11.7) is also a quasiisomorphism for such M . In particular, forany M ∈ A -bimod, we have a natural isomorphism(11.8) HH q ( P ( A ) , P ( M )) ∼ = HH q ( Q q ( A/k ) , M ) , where we identify Stab( P ( M )) ∼ = P ( M ) since P ( M ) is already additive.The target of this isomorphism is known as Mac Lane Homology and de-noted HM q ( A, M ); it has been introduced by Mac Lane [Mc1] back in 1956.Interpretation of Mac Lane Homology in terms of bifunctor homology isthe great discovery of Jibladze and Pirashvili [JP], and the identificationbetween either of them and T HH is [FPSVW] and [PW]. The product(11.5) corresponds under (11.8) to a product induced by (11.3) and the laxmonoidal structure on the functor Q q .188 emark 11.1. As noticed in [PW], as soon one has some sufficiently well-developed “brave new algebra” that allows one to work “over the spherespectrum S ”, an identification T HH q ( A, M ) ∼ = HM q ( A, M ) becomes obvi-ous: this is simply (11.2) with k ′ = S . One advantage of working with Mac LaneHomology it that it has a cohomology theory attached to it: for any flat k -algebra A and A -bimodule M , one defines the Hochschild Cohomology HH q ( A, M ) as RHom q A o × k A ( A, M ), with the obvious generalization to DGalgebras, and then Mac Lane Cohomology is given by HM q ( A, M ) = HH q ( Q q ( A ) , M ) . Note that if A is commutative, the lax monoidal structure on Q q ( − ) and theproduct map A ⊗ k A → A turn HM q ( A ) into a commutative algebra, andthen if A = k is a field, HM q ( k ) is actually a commutative cocommutativeHopf algebra, with the dual algebra given by HM q ( k ). In this case, thebifunctor P ( k ) : P ( k ) o × P ( k ) → k -mod is isomorphic to T ∗ ⊠ k T , where T : P ( k ) → k -mod sends V ∈ P ( K ) to itself, and T ∗ : P ( k ) o → k -mod sendsit to the dual vector space V ∗ , and then (1.25) provides an isomorphism HM q ( k ) ∼ = Tor q P ( k ) ( T ∗ , T ) , where Tor q I denotes the homology modules of the derived tensor product(1.25). For cohomology, we then have HM q ( k ) ∼ = Ext q P ( k ) ( T, T ) , the Ext-groups computed in the abelian category Fun( P ( k ) , k ) (or equiva-lently, in the derived category D ( Q q ( k, k )) ∼ = Ho st ( P ( k ) , k ) ⊂ D ≥ ( P ( k ) , k )).Another advantage is computational. Since the complex computing theMac Lane Homology HM q ( A, M ) is obtained by totalizing a bicomplex, andsimilarly for HM q ( A ), we have spectral sequences(11.9) HH q ( HQ q ( A/k ) , M ) ⇒ HM q ( A, M ) ,HH q ( HQ q ( A/k ) , M ) ⇒ HM q ( A, M ) , where HQ q ( A/k ) is the homology algebra of the DG algebra Q q ( A/k ) (withzero differential). If A = M = k is a perfect field of some positive oddcharacteristic p = char k ≥ 3, then (5.1) induces an isomorphism(11.10) HQ q ( k/k ) = ( k ⊗ k )[ β, τ i , ξ i ] , E -pages of the sequences (11.9) can be described very explicitly.Namely, given some formal generators x , . . . , x n of some homological de-grees deg x , . . . , deg x n , let k { x , . . . , x n } be the free divided power algebragenerated by x q (that is, the free graded-commutative algebra generated bysymbols x ji = x [ p j ] i , j ≥ 0, 0 ≤ i ≤ n of degrees deg x ji = p j deg x i , modulothe relations ( x ji ) p = 0). Then the standard Hochschild-Kostant-Rosenbergargument provides algebra isomorphisms HH q ( k [ x , . . . , x i ] , k ) ∼ = k { y , . . . , y i } ,HH q ( k [ x , . . . , x i ] , k ) ∼ = k [ y ′ , . . . , y ′ i ] , where the free graded-commutative algebra k [ x , . . . , x n ] acts on k via theaugmentation map (that is, all x i act by 0), and y i resp. y ′ i are generatorsof homological resp. cohomological degree deg x i + 1. Plugging this into(11.10), and observing that HH q ( k ⊗ k, k ) ∼ = HH q ( k ⊗ k, k ) ∼ = k since k is perfect, we conclude that the page E q , q ( k ) resp. E q , q ( k ) of the spectralsequence (11.9) for HM q ( k ) resp. HM q ( k ) is given by(11.11) E q , q ( k ) = k { σ, A i , B i } , E q , q ( k ) = k [ ε, C i , D i ] , i ≥ , where σ , A i , B i resp. ε , C i , D i have homological resp. cohomological bide-grees (1 , p i − , p i − , p = 2, then a similar computationproduces a spectral sequence(11.12) k { σ } ⇒ HM q ( k ) , deg σ = 2that degenerates for obvious dimension reasons. For (11.11), this is not true:again for dimension reasons, the sequences degenerate up to the E p − resp. E p − -page, but then there are differentials d p − resp. d p − given by(11.13) d p − ( B [ p ] i − ) = c − i A i , d p − ( C i ) = c i D pi − for some possibily non-trivial elements c i ∈ k .As it happens, all the elements c i in (11.13) are indeed non-trivial (andthen again for dimension reasons, the spectral sequences degenerate at E p resp. E p ). For Mac Lane Homology, this was first discovered by Breen[Br], and for T HH , this is in the original paper of B¨okstedt [B¨o] (the twoarguments are completely different, and the identification between T HH and HM q was in any case only discovered later). In 1994, there appearedanother proof by Franjou, Lannes and Schwartz [FLS] using cohomologyrather than homology, and homological rather than homotopical methods.190 good overview of this proof can be found in [P1], together with a very nicedescription of the state-of-the-art at the time of its writing. Since then, therehas been a lot of progress (such as the introduction of strictly polynomialfunctors in [FS], and a lot of subsequent work in the area including thegroundbreaking paper of Touz´e [T]). With all these modern advances, andwith the ground we have prepared in Subsection 5.3, we can now cut theproof down to several lines. Let us do it for the convenience of the reader.The numbers c i in (11.13) obviously only depend on p , so that we mayassume right away that k = F p is a prime field. For any integer n ≥ 1, let P n ( k ) be the k -linear category whose objects are finite-dimensional vectorspaces, and whose morphisms are given by P n ( k )( V, V ′ ) = D n ( P ( k )( V, V ′ )) , where D n is lax monoidal divided power functor (5.21). A strictly polynomialfunctor of degree n is a k -linear functor P n ( k ) → k -mod. These form anabelian category Fun k ( P n ( k ) , k ) with its derived category D ( P n ( k )). Wewill need the following general fact. Lemma 11.2. For any n ≥ , the category Fun k ( P n ( k ) , k ) of strictly poly-nomial functors of degree n has finite homological dimension.Proof. By an easy and standard argument, for any N ≥ n , D ( P n ( k )) isequivalent to the derived category of degree- n polynomial representations ofthe algebraic group GL ( N, k ) ([BFS, Theorem 2.4], or see [E, Theorem 4.7]and the following discussion). The latter is a highest weight category andtherefore has finite homological dimension ([CPS, Theorem 3.7.2], [Do]);alternatively, by [E, Theorem 1.4], it is a full subcategory in the derivedcategory of quasicoherent sheaves on a Grassmann variety G ( N, M ) for asufficiently large M . (cid:3) Now, while the category P n ( k ) is not half-additive, we still have a functor m : Γ + × P n ( k ) → P n ( k ) induced by the functor (4.12) for the category P ( k ). We say that an object E ∈ D ( P n ( k )) is stable if so is m ∗ E ∈ D (Γ + × P n ( k ) , k ), and we denote by D st ( P n ( k )) ⊂ D ( P n ( k )) the full subcategoryspanned by stable objects. By the same argument as in Subsection 5.1, wethen have an equivalence(11.14) D st ( P n ( k )) ∼ = D ( Q n q ( k )) , where Q n q ( k ) is the DG algebra (5.23). In particular, the category is trivialunless n is a power of p . If n = p d is a power of p , we have at least one non-zero stable object given by the “iterated Frobenius” functor T ( d ) : P n ( k ) → -mod that sends a vector space to itself, and acts on morphisms via the d -fold iteration of the map (5.36). The equivalence (11.14) sends T ( d ) to thetrivial DG module k . If we let HM q ( k ; d ) be the algebra given by HM q ( k ; d ) = Ext q P n ( k ) ( T ( d ) , T ( d ) ) ∼ = Ext q Q ( d ) q ( k ) ( k, k ) , where Q ( d ) q is as in (5.40), then (5.45) provides a spectral sequence(11.15) k [ ε, C i , D i ] ⇒ HM q ( k ; d ) , where the generators are numbered by i with 1 ≤ i ≤ d and have the samedegrees as in (11.11). Moreover, these spectral sequences form an inductivesystem with respect to d , and by (5.42), the cohomological sequence of(11.11) is the colimit of the sequences (11.15) over d . In particular, theyall degenerate up to E p − , and their differentials d p − are given by (11.13),with the same numbers c i . Lemma 11.3. All the elements c i ∈ k in (11.13) are non-trivial, the spec-tral sequences (11.11) degenerate at E p resp. E p , and we have HM i ( k ) ∼ = HM i ( k ) ∼ = k if i is even and otherwise.Proof. Assume that c d = 0 for some d , take the smallest such d , and considerthe spectral sequences (11.15) for HM q ( k ; d ) and HM q ( k ; d − E p , and HM q ( k ; d − 1) is an algebraconcentrated in degrees between 0 and 2 p d − 2. But since c d = 0, this meansthat again for dimension reasons, the former also degenerates at E p . Then D d survives into the E ∞ -term and provides an element D ∈ HM q ( k ; d ) suchthat D n = 0 for any n ≥ 1. This contradicts Lemma 11.2. (cid:3) Let us now interpret Hochschild homology and itsgeneralizations as homotopy trace theories in the sense of Section 10.Hochschild homology itself is in the fact the prototypical example of atrace theory. To see it as such, consider the category k -mod fl of flat k -modules, and as in Subsection 10.3, let A lg ( k ) ⊂ M or ∗ ( k -mod fl ) be the2-category of flat k -algebras, with morphisms given by right-flat bimod-ules. As in Subsection 9.7.2, since k -mod fl is symmetric monoidal, theembedding I ( k ) : k -mod fl ⊂ k -mod has a trivial structure of a trace func-tor that we denote by I ( k ) triv ∈ Tr( Bk -mod fl , k -mod). Then for any flat k -algebra A and right-flat A -bimodule M , we have a natural identification192xp( I ( k ) triv )( h A, M i ) ∼ = Tr A ( M ). The homotopy expansion Exp ho ( I ( k ) triv )in Ho tr ( A lg ( k ) , k ) is stable, and if we denote HH ( k ) = Exp ho ( I ( k ) triv ) ∼ = Exp st ( I ( k ) triv ) ∈ Ho tr ( A lg ( k ) , k ) , then we have a functorial identification H o ( HH ( k ))( h A, M i ) ∼ = CH q ( A/k, M ) ∈ Ho( k ) ∼ = D ≤ ( k ) , where H o ( − ) is the functor (4.1). Moreover, we also have the simplicial k -module ( M/A ) ♯ = I ( k ) triv ( M/A ) ♯ ∈ ∆ o k -mod of (9.23), and its standardcomplex C q (∆ o , ( M/A ) ♯ ) gives the Hochschild complex CH q ( A, M ). Thehomotopy trace theory HH ( k ) is multiplicative, and by (9.42), the corre-sponding porduct is the usual K¨unneth product (11.3).The first non-trivial example of a trace functor is obtained by using theedgewise subdivision of Subsection 9.7.3 (it is described in some detail in[Ka3]). For any positive integer p ≥ 1, shorten notation by setting π p ( k ) = π p ( Bk -mod fl ), i p ( k ) = i p ( Bk -mod fl ), and denote I ( p ) ( k ) = π p ( k ) ∗ i p ( k ) ∗ triv I ( k ) ∈ Tr( Bk -mod fl , k -mod)the p -th edgewise subdivision of the trace functor I ( k ) triv in the sense ofDefinition 9.22. We then consider its stable homotopy expansion, and define(11.16) HH ( p ) ( k ) = Exp st ( I ( p ) ( k )) ∈ Ho sttr ( A lg ( k ) , k ) ,CH ( p ) q ( A/k, M ) = HH ( p ) ( k )( h A, M i ) ∈ Ho( k )for a flat k -algebra A and a right-flat A -bimodule M . We let HH ( p ) q ( A/k, M )be the homology modules of the complex CH ( p ) q ( A/k, M ). If M = A is thediagonal bimodule, we let CH ( p ) q ( A/k ) = CH ( p ) q ( A/k, A ), and we recall that CH ( p ) q ( A/k ) ∈ D ∀ (∆ o , k ) ∼ = D ( k ) extends to the category Λ: if we denote(11.17) CC ( p ) ( A/k ) = Adj(Exp st ( I ( p ) ( k )))( A ) ∈ D ∀ (Λ , k ) , then CH ( p ) ( A/k ) ∼ = j o ∗ CC ( p ) ( A ). Alternatively, we can first take the sta-bilization Stab( I ( p ) ( k )) ∈ Ho sttr ( Bk -mod fl , k ), and then identify HH ( p ) ( k ) =Exp ho (Stab( I ( p ) ( k ))). Note that the underlying functor k -mod fl → k -modof the trace functor I ( p ) ( k ) is the p -th cyclic power functor of (5.3), so thatStab( I ( p ) ( k )) can be computed by the same argument as in Lemma 5.1.Namely, for any bifibration π : I ′ → I of bounded categories whose fibersare connected groupoids with finite automorphisms groups, we have a nat-ural trace map tr : π ! → π ∗ between the Kan extension functors π ! , π ∗ :193un( I ′ , k ) → Fun( I, k ) (if I = pt , so that I ′ has a single object o , tr is theaveraging over the automorphism group Aut( o ), and in the general case, seee.g. [Ka11, Subsection 1.1]). Moreover, see e.g. [Ka6, Subsection 3.1], onecan define the relative Tate cohomology functor π ♭ that fits into a functorialexact triangle(11.18) R q π ∗ ( E ) −−−−→ π ♭ ( E ) −−−−→ L q π ! ( E )[1] tr −−−−→ for any E ∈ Fun( I ′ , k ), where [1] stands for the homological shift. We thenhave(11.19) Stab( I ( p ) ( k )) ∼ = τ ≥ π p ( k ) ♭ i p ( k ) ∗ I ( k ) triv . In particular, Stab( I ( p ) ( k )) fits into a distinguished triangle(11.20) I ( p ) ( k ) −−−−→ Stab( I ( p ) ( k )) −−−−→ L q π p ( k ) ! i p ( k ) ∗ I ( k )[1] −−−−→ in the triangulated category D (Λ Bk -mod fl , k ) induced by (11.18). The ob-ject CH ( p ) q ( A, M ) ∈ D ( k ) is given by CH ( p ) q ( A, M ) ∼ = C q (∆ o , τ ≥ π p ( k ) ♭ i p ( k ) ∗ ( M/A ) ♯ ) , and the homology of the category ∆ o can be again computed by takingthe standard complexes of simplicial k -modules, and totalizing the resultingbicomplex. Since I ( p ) ( k ) is the edgewise subdivision of a multiplicative tracefunctor, it is itself multiplicative, and then HH ( p ) ( k ) is a multiplicativehomotopy trace theory; the product is obtained by composing the K¨unnethproduct (11.3) and the standard lax monoidal structure on the relative Tatecohomology functor π p ( k ) ♭ .Now consider the category Sets of sets, with the cartesian product, andthe category Sets + of pointed sets, with the smash product. The forgetulfunctor F : Sets + → Sets has a left-adjoint I : Sets → Sets + sending S ∈ Setsto the coproduct S + = S ⊔ { o } ∈ Sets + , and I is symmetric monoidal, sothat F is lax monoidal by adjunction. Then again, since Sets is symmetricmonoidal, I has a trivial trace functor structure I triv ∈ Tr( B Sets , Sets + ),and we can consider its stable expansion Exp st ( I triv ) ∈ Ho sttr ( M or ∗ (Sets)).However, we also have the forgetful functor (4.32). Denote by ϕ − = F ◦ ϕ : k -mod fl → Sets its composition with the forgetful functor F , and notethat both F and ϕ are lax monoidal, so that ϕ − is lax monoidal as well.The monoidal category k -mod fl is well-generated by Example 8.37, so thatwe have the 2-functors T , P of (8.49) and (8.50), and the map (9.47) ofSubsection 9.7.2 whose stabilized homotopical counterpart reads as(11.21) Λ P ∗ Λ ϕ ∗− Λ T ∗ Exp st ( I triv ) → Exp st (( ϕ ∗− I ) triv ) . I ◦ ϕ − ∼ = ϕ ∨ pt + , where pt + is the constant functorsending everything to pt + ∈ Sets. Then Stab( pt + ) ∼ = ∅ + , the constantfunctors with value ∅ + , and the map Stab( ϕ ∗− I ) ∼ = Stab( ϕ ∨ pt + ) ∼ = Stab( ϕ ) × Stab( pt + ) → Stab( ϕ ) induced by the adjunction map I ◦ ϕ − → ϕ is anisomorphism. Combining this with (10.40), we obtain identificationsExp st (( ϕ ∗− I ) triv ) ∼ = Exp st ( ϕ ) ∼ = ϕ ◦ Exp st ( I ( k )) , so that after restriction to A lg ( k ) ⊂ M or ∗ ( k ), the target of the map (11.21)becomes naturally identified with ϕ ◦ HH ( k ). Let us denote its source by T HH ( k ) = Λ P ∗ Λ ϕ ∗− Λ T ∗ Exp st ( I triv ) | A lg ( k ) ∈ Ho sttr ( A lg ( k )) , and justify the notation by the following result. Lemma 11.4. For any flat k -algebra A and right-flat k -bimodule M , wehave a functorial identification H o st ( T HH ( k ))( h A, M i ) ∼ = T HH ( A, M ) ∼ = ϕ ( T HH q ( A, M )) ∈ Ho st (Γ + ) , where H o st ( T HH ( k )) is as in (10.12) .Proof. By definition, objects in M or ∗ (Sets) are small categories, and mor-phisms frm I to I ′ are functors F : I o × I ′ → Sets satisfying the appro-priate representability condition (namely, F ( i × − ) : I ′ → Sets is ind-polyrepresentable in the sense of Definition 8.8 for any i ∈ I ). In particular,endomorphisms of a small category I are bifunctors F : I o × I → Sets. Butwe have the functor (6.11), and then as in the proofs of Lemma 9.8 andLemma 9.5, (9.8) provides a natural identification H o (Exp ho ( I triv ))( h I, F i ) ∼ = hocolim ∆ o I I ◦ ξ ∗ ♭ ( s × t ) ∗ F for any small category I ∈ M or ∗ (Sets) [0] and its endomorphism F . Byvirtue of (4.34) and (4.36), if we apply this to I = P ( A ) and F = P ( M ), weobtain an identification H o st ( T HH ( k ))( h A, M i ) ∼ = ϕ ◦ C q (∆ o P ( A ) , ξ ∗ ♭ ( s × t ) ∗ P ( M )) , and then by (11.4) and (1.29), it only remains to prove that the functor(6.11) is homotopy cofinal. The argument for this is exactly the same as inLemma 6.11, with special maps replaced by bispecial maps. (cid:3) emark 11.5. Lemma 11.4 suggest that the unstable homotopy trace the-ory Exp ho ( I triv ) is actually more fundamental. If we take as F the identityendomorphism of a small category I , then H o (Exp ho ( I triv ))( h I, id i ) is sim-ply the geometric realization of the cyclic nerve N cy ( I ) of the category I .Considering the cyclic nerve of the category P ( A ) has been suggested byGoodwillie [Go] back in 1988; among other things, he observed that thiscyclic nerve has an infinite loop space structure induced by the direct sumin P ( A ), and gives rise to a “cyclic K -theory spectrum” of the algebra A .Morally, this should be a “master theory” that gives all other interesting the-ories by formal procedures (e.g. stabilization gives T HH ). Unfortunately,the cyclic nerve is very hard to compute; it seems that even when A = F p is a prime field, the homotopy groups of | N cy ( P ( A )) | are not known. Remark 11.6. Our notation seems to imply that T HH ( k ) somehow de-pends on k (which would be strange since T HH is an absolute theory).This an illusion: k in T HH ( k ) simply denotes the domain of definition ofthe trace theory, and for any commutative ring map f : k ′ → k , we have anatural identification f ∗ T HH ( k ) ∼ = T HH ( k ′ ).With out new notation, (11.21) is simply a map T HH ( k ) → ϕ ( HH ( k )),and under the identication of Lemma 11.4, it induces a functorial map(11.22) T HH ( A, M ) → ϕ ( CH q ( A, M ))for any flat k -algebra A and right-flat A -bimodule M . In terms of bifunctorhomology, this is the augmentation map (1.33). Moreover, T HH ( k ) is amultiplicative homotopy trace theory, and again by (9.42), the product co-incides with the functor homology product (11.5), while the augmentationmap (11.22) is a multiplicative map.In addition, since the product in Sets is cartesian, a natural map (9.50)for the identity functor Id and any integer l is given by the diagonal maps S → S l , S ∈ Sets, and this then induces a map (9.50) for any functor E : Sets → E to some category E . In particular, this applies to I , so thatthe expansion Exp ho ( I triv ) and its stabilization Exp st ( I triv ) acquire a natural h F, l i -structure in the sense of Definition 9.21. The map (11.21) then inducesa natural map(11.23) T HH ( k ) → ϕ ◦ HH ( p ) ( k ) , where HH ( p ) ( k ) is the homotopy trace theory (11.16). For any flat k -algebra196 and right-flat A -bimodule M , we then have functorial maps(11.24) ψ : T HH ( A, M ) → ϕ ( CH ( p ) q ( A/k, M )) ,ψ q : T HH q ( A, M ) → HH ( p ) q ( A/k, M ) . Here ψ is a map in Ho st (Γ + ) (that is, a map of connective spectra), and ψ q is the induced map on homotopy groups. Since (11.23) is a map of tracetheories, ψ commutes with the corresponding maps B of (10.19). Moreover,(11.23) is a map of multiplicative trace theories, so that in the case M = A = k , ψ and ψ q are multiplicative maps. However, neither of them needsto be k -linear (and in interesting example, they are not). Now fix a prime p , and from now on, assumethat k is a perfect field of characteristic p . Then T HH q ( k ) ∼ = HM q ( k ) isthe abutment of the spectral sequence (11.9) with the E -page (11.11), or(11.12) if p = 2, and in either case, the sequence for T HH i ( k ), i = 0 , , T HH ( k ) ∼ = k , T HH ( k ) = 0,and T HH ( k ) ∼ = k is the free k -module generated by a single element σ .Morover, T HH q ( k ) is a commutative cocomutative Hopf algebra, and inparticular, we have a natural algebra map(11.25) α : k [ σ ] → T HH q ( k ) , where k [ σ ] is the free commutative k -algebra generated by σ , deg σ = 2. Proposition 11.7. The map (11.25) is an isomorphism. Our proof of Proposition 11.7 uses the maps (11.24) — specifically, wetake A = M = k , and consider the corresponding maps(11.26) ψ : T HH ( k ) → ϕ ( CH ( p ) q ( k/k )) , ψ q : T HH q ( k ) → HH ( p ) q ( k/k ) . Since the underlying functor k -mod fl → k -mod of the trace functor I ( p ) ( k )is the cyclic power functor C of (5.3), for any M ∈ k -mod fl , we have(11.27) CH ( p ) q ( k/k, M ) ∼ = R ( M ) ∼ = CH q ( k/k, R ( M )) ∼ = ∼ = CH q ( P ( k ) /k, P ( M ) ∗ R ) , where R ∈ Ho( k -mod , k ) is the stabilization (5.4) of the functor C . Themap (11.26) is then obtained by stabilizing the map hocolim tw ( P ( k )) ϕ ( s × t ) ∗ P ( M )) → hocolim tw ( P ( k )) ϕ (( s × t ) ∗ P ( M ) ∗ C )197nduced by (5.6), taking M = k , and composing the resulting map T HH ( k ) ∼ = ϕ ( CH q ( P ( k ) , P ( k ))) → ϕ ( CH q ( P ( k ) , P ( k ) ∗ R ))with the map (1.33) for the category P ( k ). In particular, since the map(5.6) is obviously Frobenius-semilinear, so are the maps ψ q of (11.26): wehave ψ i ( λa ) = λ p ϕ i ( a ) for any i ≥ λ ∈ k and a ∈ T HH i ( k ).Now recall that we also have the object CC ( p ) q ( k/k ) of (11.17) such that CH ( p ) q ( k/k ) ∼ = j o ∗ CC ( p ) q ( k/k ) ∈ D ∀ (∆ o , k ) ∼ = D ( k ). By (11.19), it is givenby CC ( p ) q ( k/k ) ∼ = τ ≥ π ♭p k , where k is the constant functor with value k .Then (11.20) induces an exact triangle(11.28) k −−−−→ CC ( p ) q ( k/k ) −−−−→ L q π p ! k −−−−→ in D ♮ (Λ , k ) that restricts to a triangle(11.29) k −−−−→ R ( k ) r −−−−→ R ( k ) ∼ = C q ( Z /p Z , k ) −−−−→ in D ∀ (∆ o )( k ) ∼ = D ( k ) (where R ( k ) ∼ = CH ( p ) q ( k/k ) by (11.27)). All terms in(11.29) carry the Connes-Tsygan differential (10.19), and the differential for R ( k ) is easy to compute: it has been shown in [Ka4, Lemma 3.2] that(11.30) L q π p ! k ∼ = M i ≥ K ( k )[2 i ] , where K ( k ) ∼ = K q ⊗ k is the complex (10.10), and then H i ( Z /p Z , k ) ∼ = k inany non-negative degree i ≥ 0, and B = B i : H i ( Z /p Z , k ) → H i +1 ( Z /p Z , k )is given by(11.31) B i = ( id , i = 2 j, , i = 2 j + 1 . As an algebra, HH ( p ) q ( k/k ) ∼ = τ ≤ ˇ H q ( Z /p Z , k ) is the free graded-commuta-tive algebra given by(11.32) HH ( p ) q ( k/k ) ∼ = k [ β, σ ] , deg σ = 2 , deg β = 1if p is odd. If p = 2, then graded-commutative means commutative, and wehave ε = σ instead of ε = 0. Lemma 11.8. The component ψ : T HH ( k ) → HH ( p )2 ( k ) of the map (11.26) is an isomorphism. roof. Since T HH ( k ) = 0, the map (10.19) for T HH ( k ) factors throughthe canonical truncation τ ≥ Ω T HH ( k ) ∼ = Ω τ ≥ T HH ( k ), and by (11.31),the same is true for R ( k )[1]. Applying further the canonical truncation τ ≤ ,we obtain a diagram ϕ ( k ) ∼ = τ ≤ T HH ( k ) B −−−−→ ϕ ( k [1]) ∼ = τ ≤ τ ≥ Ω T HH ( k ) ψ ′ y y ψ ϕ ( k [1]) ∼ = τ ≤ ϕ ( R ( k )[1]) B −−−−→ ϕ ( k [1]) ∼ = τ ≤ τ ≥ ϕ ( R ( k ))in Ho st (Γ + ), where ψ ′ = r ◦ ψ is the composition of ψ of (11.26) and theprojection r in (11.29). If we further compose everything with the embed-ding τ ≤ τ ≥ ψ ( R ( k )) → τ ≤ ψ ( R ( k )), the diagram becomes commutative, andsince the embedding is split, it was commutative to begin with. But B inthe bottom line is an isomorphism, and ψ ′ is not equal to 0 by Lemma 5.2.Therefore ψ : k → k is not equal to 0 either; being Frobenius-semilinear, itmust be an isomorphism. (cid:3) Proof of Proposition 11.7. Lemma 11.8 and (11.32) immediately show thatthe map (11.25) is injective: ϕ n α ( σ n ) = ϕ ( σ ) n = 0 for any n , so α ( σ n ) = 0as well. If p = 2, then the degenerate spectral sequence (11.12) showsthat the source and the target of (11.25) are k -vector spaces of the samedimension, so this finishes the proof. If p ≥ (cid:3) Remark 11.9. The original proof of Proposition 11.7 in [B¨o] used non-trivial techniques from homotopy theory such as Dyer-Lashof operations.Very recently, a very short topological proof appeared in [KN] where theauthors deduce the theorem from a more fundamental fact: Q q ( k ) consideredas an E -algebra is free on one generator β of homological degree 1. Thededuction is extremely easy but the fact itself again requires Dyer-Lashofoperations and other homotopy theory techniques (including some classicalbut non-trivial computations). It would be interesting to see if one can finda direct proof in terms of the description of Q q ( k ) given in Subsection 5.4. Proposition 11.7 shows that if k isa perfect field of characteristic p ≥ 0, then the map ψ q of (11.26) is injec-tive. It is certainly not an isomorphism, since its target is k in all degreesincluding the odd ones. It turns out that one can modify the target so thatthe map becomes an isomorphism, thus giving a purely algebraic model for199 HH ( A, M ). This uses Hochschild-Witt Homology of [Ka11] and polyno-mial Witt vectors of [Ka8]. Let us briefly recall the construction.Recall that for any integer m ≥ 1, we have the ring W m ( k ) of m -truncated p -typical Witt vectors, with the restriction and Frobenius ringmaps R, F : W m +1 ( k ) → W m ( k ). If m = 1, then W ( k ) ∼ = k , and the map F : W ( k ) → k is the composition of the map R and the Frobenius endomor-phism of k , and since k is assumed to be perfect, the ideal Ker R ⊂ W m ( k )for any m is generated by p . Now for any m ≥ 0, simplify notation bywriting i p m = i p m ( W m +1 ( k )), π p m = π p m ( W m +1 ( k )), I triv = I ( W m +1 ( k )) triv ,and define trace functors Q m , Q ′ m ∈ Tr( BW m +1 ( k )-mod fl , W m +1 ( k )) and amap R : Q ′ m → Q m by constructing the diagram(11.33) π p m ! i ∗ p m I triv p tr −−−−→ π p m ∗ i ∗ p m I triv −−−−→ Q ′ m −−−−→ p id y id y y R π p m ! i ∗ p m I triv tr −−−−→ π p m ∗ i ∗ p m I triv −−−−→ Q m −−−−→ m = 0, we have Q = 0 and Q ′ o = I triv . Asit turns out ([Ka8, Proposition 2.3]), for any m ≥ W m +1 ( k )-module M , Q m ( M ) and Q ′ m ( M ) only depend on the quotient M/p , so thatwe have Q m ∼ = q ∗ W m , Q ′ m = q ∗ W ′ m for some trace functors W m , W ′ m ∈ Tr( Bk -mod , W m ( k )), where q : W m +1 ( k )-mod fl → k -mod sends M to M/p .One further shows ([Ka8, Corollary 2.7]) that W m +1 ∼ = R ∗ W m (where as inExample 5.5, for any ring map f : k ′ → k , we denote by f ∗ : k -mod → k -modthe restriction of scalars). Therefore W ( M ) ∼ = M , and the map R in (11.33)induces a functorial restriction map R : W m +1 → R ∗ W m . All the functors W m are also symmetric lax monoidal ([Ka8, Proposition 3.10]), and themaps R are lax monoidal, so that the limit W = lim W q is a symmetric laxmonoidal trace functor on k -mod with values in p -adically complete mod-ules over W ( k ) = lim W q ( k ). Moreover, the Frobenius maps F : W m +1 ( k ) → W m ( k ) lift to lax monoidal trace functor maps F : W m +1 → F ∗ W m , and wealso have trace functor maps C : R ∗ W m → W m +1 that fit into short exactsequences(11.34) 0 −−−−→ R m ∗ W n C m −−−−→ W m + n F n −−−−→ F n ∗ π p m ! i ∗ p m W m −−−−→ m, n ≥ 0. We recall the functor W already appeared in the proofof Lemma 11.3, and (11.34) for m = n = 1 is (5.13). The Frobenius maps F commute with the restriction maps R , thus define an h F, p i -structure inthe sense of Definition 9.21 on the limit trace functor W , and the map(11.35) F : W → π p ∗ i ∗ p W W st = Stab( W ) of the trace functor W . Namely, for any k -vector space M , W ( M ⊗ k p ) carries a natural Z /p Z -action induced by thetrace functor structure on W , with the Z /p Z -equivariant map(11.36) W ( M ⊗ k p ) → W ( M ⊗ k p ) ∼ = M ⊗ k p induced by the restriction map W → W , and we have the following. Lemma 11.10. The map ˇ H i ( Z /p Z , W ( M ⊗ k p )) → ˇ H i ( Z /p Z , M ⊗ k p ) induc-ed by (11.36) is bijective if i is even, and its source vanishes if i is odd.Proof. This is [Ka8, Corollary 3.9]. (cid:3) By the same argument as in Lemma 5.1, Lemma 11.10 together with theisomorphism (11.35) provide an identification W st ∼ = τ ≥ π p ( k ) ♭ i p ( k ) ∗ W, where π ♭ is as in (11.19), so that W st fits into an exact triangle(11.37) W −−−−→ W st −−−−→ L q π p ! i ∗ p W [1] −−−−→ in the category Ho tr ( Bk -mod , W ( k )). As in (11.16), we can now take thestable homotopy expansion, and define(11.38) WHH st = Exp st ( W ) ∼ = Exp( W st ) ∈ Ho sttr ( A lg ( k ) , W ( k )) ,W CH st q ( A/k, M ) = WHH ( h A, M i ) ∈ Ho( W ( k ))for a flat k -algebra A and a right-flat A -bimodule M . By (9.27) and (9.24),we have(11.39) W CH st q ( A/k, M ) ∼ = C q (∆ o , W st ( M/A ) ♯ ) ∈ D ( W ( k ))for a functorial object W st ( M/A ) ♯ ∈ Ho(∆ o , W ( k )), and we denote by W HH st ( A/k, M ) the homology modules of the object (11.39). If M = A is the diagonal bimodule, then W st ( A/A ) ♯ = j o ∗ W st ( A ) ♯ for a cyclic object W ( A ) ♯ ∈ Ho(Λ , W ( k )), and this is again functorial with respect to A in thesense of Lemma 9.14.If we do not stabilize, the same procedure gives a homotopy trace the-ory WHH = Exp( W ) ∈ Ho tr ( A lg ( k ) , W ( k ), functorial objects W ( M/A ) ♯ ∈ Fun(∆ o , W ( k )), W ( A ) ♯ ∈ Fun(Λ o , W ( k )), and then the Hochschild-Witt Ho-mology modules W HH q ( A, M ) the Hochschild-Witt complex W CH q ( A, M )201ntroduced and studied in some detail in [Ka11]. We also have the functorialstabilization map(11.40) W HH q ( A, M ) → W HH st q ( A, M )induced by a map of homotopy trace theories WHH → WHH st . By (11.37),the corresponding map W ( M/A ) ♯ → W st ( M/A ) ♯ fits into an exact triangle(11.41) W ( M/A ) ♯ −→ W st ( M/A ) ♯ −→ L q π p ! i ∗ p W ( M/A ) ♯ [1] −→ in D (∆ o , W ( k )), and if M = A is the diagonal bimodule, the triangle comesfrom a triangle in D (Λ , W ( k )). Moreover, both WHH and WHH st aremultiplicative homotopy trace theories, so that we have product maps W HH q ( A, M ) ⊗ W ( k ) W HH q ( A ′ , M ′ ) → W HH q ( A ⊗ k A ′ , M ⊗ k M ′ ) W HH st q ( A, M ) ⊗ W ( k ) W HH st q ( A ′ , M ′ ) → W HH st q ( A ⊗ k A ′ , M ⊗ k M ′ ) , and the map (11.40) is compatible with these maps. In addition to this, therestriction map W ( M ) → W ( M ) ∼ = M induces a map of homotopy tracetheories r : WHH → HH ( k ) that factors through (11.40) since HH ( k ) isstable, and by virtue of the isomorphism (11.35), its p -fold edgewise subdi-vision defines a multiplicative map(11.42) WHH st → HH ( p ) ( k ) , where HH ( p ) ( k ) is as in (11.16).To compare W HH st ( A, M ) to Topological Hochschild Homology, weneed one more ingredient introduced in [Ka8, Subsection 3.3]: the non-additive Teichm¨uller map T : ϕ ( M ) → ϕ ( W m ( M )). It is induced by themap N → Q m ( N ), N ∈ W m ( k ) sending n ∈ N to n ⊗ p m , and it in factfactors through M = N/p ([Ka8, Lemma 2.2]). We have ϕ ( R ) ◦ T = T ,so that the individual Teichm¨uller maps combine together to a map T : ϕ ( M ) → ϕ ( W ( M )). Moreover, this is a map of symmetric lax monoidaltrace functors, thus gives rise to a map (9.47) that in this case reads as amap(11.43) T : T HH ( k ) → ϕ ( WHH st )of multiplicative stable homotopy trace theories on k -mod. Thus for anyflat k -algebra A and k -flat A -bimodule M , we have a map(11.44) T HH q ( A, M ) → W HH st ( A, M ) , and these maps are functorial in M and compatible with the products.202 roposition 11.11. The map (11.43) is an isomorphism, so that the map (11.44) is an isomorphism for any A and M .Proof. By the universal property of expansion, it suffices to check that themap(11.45) T q : T HH q ( k ) → W HH st ( k )induced by (11.43) is an isomorphism. Indeed, (11.42) induces a map S q : W HH st ( k ) → HH ( p ) ( k )whose composition S q ◦ T q with (11.45) is the map ψ q of (11.26). But byProposition 11.7 and Lemma 11.10, both S q and ψ q = S q ◦ T q are injectiveand have the same image. (cid:3) Proposition 11.11 shows that for algebras over aperfect field k of positive characteristic p , Topological Hochschild Homologyis W ( k )-linear as a homotopy trace theory: we have T HH ( k ) ∼ = ϕ ( WHH st ),and WHH st is W ( k )-linear by construction. In particular, the map B of(10.19) for T HH ( k ) is W ( k )-linear. This allows us to prove a comparisontheorem for Periodic Topological Cyclic Homology T P ( A ) introduced in [H4].In general, this is defined by promoting T HH ( A ) to a functor with valuesin Ho sttr (Γ + × Λ), as in (10.5), refining (10.9) to interpret the latter as thecategory of connective spectra equipped with an action of the circle S , andtaking the Tate fixed points with respect to the circle. Tate fixed pointsnormally take one out of the world of connective spectra, so this is beyondthe scope of this paper. However, there is an alternative construction thatworks perfectly well in the homological setting. The cohomology H q (Λ , Z ) =Ext q Λ ( Z , Z ) of the category Λ with integer coefficients is the algebra Z [ u ] witha generator u of degree 2 known as the “inverse Bott periodicity generator”.Explicitly, the generator is represented by the complex (10.10). For any ring R and object E ∈ Fun(Λ , R ), the homology H q (Λ , R ) is naturally a moduleover H q (Λ , Z ), and one can consider the object(11.46) CP q ( E ) = R q lim n C q (Λ , E )[ − n ] ∈ D ( R ) , where the limit is taken with respect to the inverse Bott periodicity maps u . Its homology objects are denoted HP q ( E ).For the usual Hochschild Homology, HP q ( A/k ) = HP q ( A ♯ ) is the pe-riodic cyclic homology of a flat k -algebra A , and for the Hoshchild-Witt203omology, the periodic theory W HP q ( A ) = HP q ( W ( A ) ♯ ) has been intro-duced and studied in [Ka11]. We can also consider the stabilized theory(11.47) W HP st q ( A ) = HP q ( W st ( A ) ♯ ) . The isomorphism (11.43) of Proposition 11.11 then also induces an isomor-phism of the corresponding functors of Lemma 9.14, and since the trace the-ory WHH st is obviously homotopy invariant in the sense of Subsection 10.3,the functors are defined on the whole category Alg( k ) of k -algebras and non-unital maps. Therefore (11.47) is functorially isomorphic to the TopologicalPeriodic Cyclic Homology groups T P q ( A ) of [H4], and (11.40) induces afunctorial map(11.48) W HP q ( A ) → T P q ( A ) ∼ = W HP st q ( A ) . If A is commutative and finitely generated over k , and X = Spec A is smooth,then it has been proved in [Ka11, Theorem 6.15] that the Hochschild-WittHomology groups W HH q ( A ) are naturally identified with the spaces of deRham-Witt forms on X , and the differential (10.19) corresponds to thedifferential in the de Rham-Witt complex. It relatively straightforward todeduce from this that if p is large enough, or if one inverts p , then W HP q ( A )is the 2-periodized version of the cristalline cohomology H q cris ( X ). For the T P -theory, analogous results were established in [H4], and this seems con-tradictory since (11.40) is not an isomorphism. However, it becomes one ifwe pass to the periodic versions. Lemma 11.12. For any perfect field k and k -algebra A , the map (11.48) isan isomorphism.Proof. Since W ( M ) is p -adically complete for any k -vector space M , theobject W ( A ) ♯ ∈ Fun(Λ , W ( k )) is also p -adically complete, and so is itsrestriction i ∗ p W ( A ) ♯ ∈ Fun(Λ p , W ( k )). Then by (11.41), it suffices to showthat CP q ( L q π p ! E ) = 0 for any p -adically complete E ∈ Fun(Λ p , E ). Indeed,we have C q (Λ , L q π p ! E ) ∼ = C q (Λ p , E ) , and the transition maps in the inverse system (11.46) for CP q ( L q π p ! E ) aregiven by the action of the element π ∗ p u ∈ H (Λ p , Z ). But π ∗ p u = pi ∗ p u , sothis element is divisible by p , and to finish the proof, it suffices to check that C q (Λ p , E ) is p -adically complete. Indeed, it can be computed by (1.28), andit is well-known (see e.g. [FT, Appendix, A3]) that k ∈ Fun(Λ o , k ) admits aprojective resolution P q whose every term is the sum of a finite number of204epresentable functors. Therefore as in Remark 4.28, C q (Λ op , − ) commuteswith arbitrary products in D ≤ (Λ p , k ) and sends p -adically complete objectsto p -adically complete ones. (cid:3) Another useful comparison result suggested to us by A. Mathew ex-presses Topological Hochschild Homology groups T HH q ( A ) in terms of theso-called co-periodic cyclic homology HP q ( A/k ) introduced in [Ka7]. Todefine it, recall that for any commutative ring k and flat k -algebra A , thatConnes-Tsygan differential B of (10.19) can be represented by an explicitfunctorial endomorphism of the standard Hochschild complex CH q ( A/k ),and then the periodic cyclic homology HP q ( A/k ) can be computed by thecomplex CP q ( A/k ) = CH q ( A/k )(( u )) with the differential b + uB , where b isthe differential in the Hochschild complex, and u is a generator of cohomo-logical degree 2. Then HP q ( A ) is defined as the homology of the complex CP q ( A/k ) = CH q ( A/k )(( u − )), with the same differential. If k contains Q ,these groups vanish taulologically, but in other cases they are non-trivial andprovide an interesting homological invariant. In particular, if k is a perfectfield of positive characteristic p , then it has been shown in [Ka7, Proposition4.4] that we have a functorial isomorphism(11.49) CP q ( A ) ∼ = C q (Λ , π ♭p i ∗ p A ♯ [1])in the derived category D ( k ). For any object E ∈ Fun(Λ p , k ), the relativeTate cohomology complex π ♭p E is equipped with an isomorphism u : π ♭p E → π ♭p E [ − 2] that corresponds to the generator u ∈ H ( Z /p Z , k ), and in termsof (11.49), this corresponds to the k (( u − ))-module structure on CP q ( A/k ).Moreover, π ♭p E [1] is equipped with the increasing conjugate filtration V q , V i π ♭p E [1] = τ ≤ i ( π ♭p E [1]), it induces a generalized conjugate filtration(11.50) V i CP q ( A ) ∼ = C q (Λ , V i π ♭p i ∗ p A ♯ [1])on CP q ( A ), and by [Ka7, Proposition 6.4], this gives rise to a convergent conjugate spectral sequence HH q ( A/k )(( u − )) ⇒ HP q ( A ).Now consider the Hochschild-Witt trace theory and the correspondingobject W ( A ) ♯ ∈ Fun(Λ , W ( k )). Then the restriction maps R : W ( M ) → M , M ∈ k -mod induces a functorial W ( k )-linear map R : W ( A ) ♯ → R ∗ A ♯ , andthis gives rise to a functorial W ( k )-linear map(11.51) W st ( A ) ♯ [1] ∼ = τ ≤ π ♭p W ( A ) ♯ [1] → V π ♭p i ∗ p R ∗ A ♯ [1] . Moreover, relative Tate cohomology is a lax monoidal functor, so that forany E ∈ Fun(Λ p , k ), we have a natural product map π ♭p k ⊗ k π ♭p E → π ♭p E .205y (11.30), we have π ♭p k ∼ = K ( k )[ − u )), so that in particular, we have aproduct map K ( k )[ − ⊗ k π ♭p E → π ♭p E . This allows to refine (11.51) to afunctorial W ( k )-linear map(11.52) K q ⊗ W st ( A ) ♯ → V π ♭p i ∗ p R ∗ A ♯ [1] , where K q is the complex (10.10). Lemma 11.13. The map (11.52) is an isomorphism in D (Λ , W ( k )) .Proof. For any n ≥ 1, let e n : pt → Λ be the embedding onto [ n ] ∈ Λ. Itsuffices to check that (11.52) becomes a quasiisomorphism after applying e ∗ n for all n . We have e ∗ n π ♭p i ∗ p A ♯ ∼ = ˇ C q ( Z /p Z , M ⊗ p ) , e ∗ n π ♭p i ∗ p W ( A ) ♯ ∼ = ˇ C q ( Z /p Z , W ( M ⊗ p )) , where M = A ⊗ n , and the Z /p Z -action on W ( M ⊗ p ) is twisted as in [Ka8,Corollary 3.9]. As we saw in Lemma 5.1, ˇ H i ( Z /p Z , M ⊗ p ) ∼ = M (1) for anyinteger i , while by [Ka8, Corollary 3.9], ˇ H i ( Z /p Z , W ( M ⊗ p )) ∼ = M (1) if i iseven and 0 if i is odd. The map (11.52) is an isomorphism in homologicaldegree 0, and since it commutes with u , it suffices to check that for any map,the map(11.53) ˇ H ( Z /p Z , M ⊗ p ) → ˇ H ( Z /p Z , M ⊗ p )given by the action of the generator ε ∈ H ( Z /p Z , k ) ∼ = k is an isomorphism.But for M = k this is true tautologically, and both the source and the targetof (11.53) commute with arbitrary sums. (cid:3) Remark 11.14. The main argument in the proof of Lemma 11.13 is essen-tially [Ka7, Proposition 5.6] but there is one difference: in [Ka7], we usedrelative Tate homology instead of relative Tate cohomology, and the coac-tion map rather than the action map. The difference between Tate homologyand Tate cohomology is simply the homological shift [1], and if p = 2, botharguments work, but the homological argument requires the equality ε = 0that is not true for p = 2. The cohomological argument works for all p (andit can be used to improve [Ka7] obviating the need for [Ka6, Section 4]). Corollary 11.15. There exists a functorial isomorphism (11.54) T HH ( A ) ∼ = V CP q ( A ) ∈ D ( k ) , where V is the conjugate filtration (11.50)206 roof. Apply C q (Λ , − ) to the isomorphism (11.52), and recall that by (10.8),we have C q (∆ o , j o ∗ E ) ∼ = C q (Λ , K ⊗ E ) for any E ∈ D (Λ , k ). (cid:3) We note that T HH ( A ) is a module over T HH ( k ) ∼ = k [ σ ], and byLemma 11.3, possibly after rescaling by a non-zero constant, the isomor-phism (11.54) intertwines the B¨okstedt periodicity generator σ with theBott periodicity generator u − . Therefore we also have T HH ( A ) ⊗ k [ σ ] k ( σ, σ − ) ∼ = CP q ( A ) , and the conjugate spectral sequence corresponds to the spectral sequenceconnecting T HH q ( A ) and HH q ( A )[ σ ]. Remark 11.16. Co-periodic cyclic homology is a non-commutative coun-terpart of the non-standard derived de Rham cohomology obtained by con-sidering the de Rham complex of a free simplicial resolution of a commu-tative algebra, and then taking its “wrong” totalization. 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